This is the easy version of this problem. The only difference is the limit of n - the length of the input string. In this version, 1 ≤ n ≤ 2000. The hard version of this challenge is not offered in the round for the second division. 

Let's define a correct bracket sequence and its depth as follow:

  * An empty string is a correct bracket sequence with depth 0. 
  * If "s" is a correct bracket sequence with depth d then "(s)" is a correct bracket sequence with depth d + 1. 
  * If "s" and "t" are both correct bracket sequences then their concatenation "st" is a correct bracket sequence with depth equal to the maximum depth of s and t. 



For a (not necessarily correct) bracket sequence s, we define its depth as the maximum depth of any correct bracket sequence induced by removing some characters from s (possibly zero). For example: the bracket sequence s = "())(())" has depth 2, because by removing the third character we obtain a correct bracket sequence "()(())" with depth 2.

Given a string a consists of only characters '(', ')' and '?'. Consider all (not necessarily correct) bracket sequences obtained by replacing all characters '?' in a by either '(' or ')'. Calculate the sum of all the depths of all these bracket sequences. As this number can be large, find it modulo 998244353.

Hacks in this problem in the first division can be done only if easy and hard versions of this problem was solved.

Input

The only line contains a non-empty string consist of only '(', ')' and '?'. The length of the string is at most 2000.

Output

Print the answer modulo 998244353 in a single line.

Examples

Input


??


Output


1


Input


(?(?))


Output


9

Note

In the first test case, we can obtain 4 bracket sequences by replacing all characters '?' with either '(' or ')':

  * "((". Its depth is 0; 
  * "))". Its depth is 0; 
  * ")(". Its depth is 0; 
  * "()". Its depth is 1. 



So, the answer is 1 = 0 + 0 + 0 + 1.

In the second test case, we can obtain 4 bracket sequences by replacing all characters '?' with either '(' or ')':

  * "(((())". Its depth is 2; 
  * "()()))". Its depth is 2; 
  * "((()))". Its depth is 3; 
  * "()(())". Its depth is 2. 



So, the answer is 9 = 2 + 2 + 3 + 2.
import java.util.*;
import java.io.*;
import java.io.FileWriter; 
import java.math.BigInteger;

// Solution

public class Main 
{    
  public static void main (String[] argv) 
  {
	  new Main();
  } 
     
  
  boolean test = false;  
  final int[][] dirs8 = {{0,1}, {0,-1}, {-1,0}, {1,0}, {1,1},{1,-1},{-1,1},{-1,-1}};
  final int[][] dirs4 = {{0,1}, {0,-1}, {-1,0}, {1,0}};
  final int MOD = 998244353;//1000000007;  
  int n_bit = 1;
  int[] bit;
  
  
  final int WALL = -1;
  final int EMPTY = -2;
  final int VISITED = 1;
  final int FULL = 2;
  final char L = '(';
  final char R = ')';
  final char MK = '?';
   
  long[] fac = new long[2001];
  long[] ifac = new long[2001];
  long[][] nchoosekSum = new long[2001][2001];
  public Main()   {
	FastReader in = new FastReader(new BufferedReader(new InputStreamReader(System.in)));
	
     
    //FastReader in = new FastReader(new BufferedReader(new FileReader("Main.in")));   
    
    //int nt = in.nextInt();    
    int nt = 1;
    StringBuilder sb = new StringBuilder();
    
    fac[0] = 1;
    ifac[0] = 1;
    for (int i = 1; i <= 2000; i++) {
        fac[i] = fac[i-1] * i % MOD;
        ifac[i] = ifac[i-1] * inv(i) % MOD;
    }
    
    for (int i = 0; i <= 2000; i++)
    {
        nchoosekSum[i][0] = 1;
        for (int j = 1; j <= i; j++)
        {
            long cur = fac[i] * ifac[j] % MOD * ifac[i-j] % MOD;
            nchoosekSum[i][j] = (nchoosekSum[i][j-1] + cur) % MOD;            
        }
    }
    
    for (int it = 0; it < nt; it++) 
    {
        
       char[] a = in.next().toCharArray();
       int n = a.length;
       
       int[] nleft = new int[n+2];
       int[] nright = new int[n+2];
       int[] nmark = new int[n+2];
       for (int i = 1; i <= n; i++) {
           nleft[i] = nleft[i-1];
           nmark[i] = nmark[i-1];
           
           if (a[i-1] == L) {
               nleft[i]++;
           }else if (a[i-1] == MK) {
               nmark[i]++;
           }
       }
       for (int i = n; i > 0; i--)
       {
           nright[i] = nright[i+1];
           if (a[i-1] == R) nright[i]++;
       }
       
       int MAX = n/2;
       long[] dp = new long[MAX+2];
       for (int i = 1; i <= MAX; i++)
       {
           for (int j = i; j + i <= n; j++)
           {
   //System.out.println("dp " + i + ": try  " + j); 
               if (a[j-1] != R && nleft[j] + nmark[j] >= i && nright[j+1] + nmark[n] - nmark[j] >= i){
                 int leftFixed = i - nleft[j];
                 int rightFixed = i - nright[j+1];
                 if (a[j-1] != L && leftFixed <= 0) break;
                 
                 if (a[j-1] == L) {
                    if (leftFixed >= 0) {
                      dp[i] = (dp[i] + choose(nmark[j], leftFixed) * chooseSum(nmark[n] - nmark[j], rightFixed) % MOD) % MOD;        
                    }
                 }else {
                    // mark this one as L
                    leftFixed--;
                    dp[i] = (dp[i] + choose(nmark[j]-1, leftFixed) * chooseSum(nmark[n] - nmark[j], rightFixed) % MOD) % MOD; 
                 }                 
               }               
           }
           //System.out.println("dp: " + i + " " + dp[i]);           
       }
       
       long tot = 0;
       for (int i = MAX; i >= 1; i--)
       {
           tot = (tot + i * (dp[i]  - dp[i+1]) % MOD) % MOD;
           tot = (tot + MOD) % MOD;
       }
       sb.append(tot+"\n");
       //sb.append(f(n) + "\n");
       //System.out.println();
       //String ans = new String(a);
       //sb.append(ans);
    }
    System.out.print(sb);
  }
  
  private long chooseSum(int n, int k){
      //System.out.println("chooseRight " + n + " " + k);
      if (k <= 0) 
        return nchoosekSum[n][n];
      return ((nchoosekSum[n][n] - nchoosekSum[n][k-1]) % MOD + MOD) % MOD;
  }
  private long choose(int n, int k)
  {
 // System.out.println("chooseLeft " + n + " " + k);
      return fac[n] * ifac[k] % MOD * ifac[n-k] % MOD;
  }
  
  
  class DSU {
      int n;
      int[] par;
      int[] sz;
      int nGroup;
      
      public DSU(int n)
      {
          this.n = n;
          par = new int[n];
          sz = new int[n];
          for (int i = 0;  i < n; i++){
              par[i] = i;
              sz[i] = 1;
          }
          nGroup = n;          
      }
      
      private boolean add(int p, int q) {
          int rp = find(p);
          int rq = find(q);
          if (rq == rp) return false;
          
          if (sz[rp] <= sz[rq]) {
              sz[rq] += sz[rp];
              par[rp] = rq;
          }else {
              sz[rp] += sz[rq];
              par[rq] = rp;
          }
          nGroup--;
          return true;
      }
      
      private int find(int p)
      {
          int r = p;
          while (par[r] != r) r = par[r];
          
          while (r != p) {
              int t = par[p];
              par[p] = r;
              p = t;
          }
          return r;
      }
      
  }
  
  private int[] build_z_function(String s)
  {
      int n = s.length();
      int[] zfun = new int[n];
      
      int l = -1, r = -1;
      for (int i = 1; i < n; i++)
      {
         // Set the start value
         if (i <= r)
            zfun[i] = Math.min(zfun[i-l], r - i + 1);
         
         while (i + zfun[i] < n && s.charAt(i + zfun[i]) == s.charAt(zfun[i])) 
             zfun[i]++;             
         
         if (i + zfun[i] - 1> r){
             l = i;
             r = i + zfun[i] - 1;
         }
      }
      
      if (test)
      {
          System.out.println("Z-function of " + s);
          for (int i = 0; i < n; i++) System.out.print(zfun[i] + " ");
          System.out.println();
      }
      return zfun;
  }
  
  private int query(int p)
  {
      int sum = 0;
      for (; p > 0; p -= (p & (-p)))
      {
          sum += bit[p];
      }
      return sum;
  }
  
  private void add(int p, int val)
  {
      //System.out.println("add to BIT " + p);
      for (; p <= n_bit; p += (p & (-p)))
      {
          bit[p] += val;
      }
  }
  
  
  
  private List<Integer> getMinFactor(int sum)
  {
      List<Integer>  factors = new ArrayList<>();
      
               for (int sz = 2; sz <= sum / sz; sz++){
                   if (sum % sz == 0) {
                       factors.add(sz);
                       factors.add(sum / sz);
                   }
               }
               if (factors.size() == 0)
                   factors.add(sum);
          
      return factors;
  }
  
  /* Tree class */
  class Tree {
      int V = 0;
      int root = 0;
      List<Integer>[] nbs;
      int[][] timeRange;
      int[] subTreeSize;
      int t;
      boolean dump = false;
      
      public Tree(int V, int root)
      {
          if (dump) 
              System.out.println("build tree with size = " + V + ", root = " + root); 
          
          this.V = V;          
          this.root = root;
          nbs = new List[V];
          subTreeSize = new int[V];
          for (int i = 0; i < V; i++)
              nbs[i] = new ArrayList<>();
          
              
      }
      
      public void doneInput()
      {
          dfsEuler();
      }
      
      public void addEdge(int p, int q)
      {
          nbs[p].add(q);
          nbs[q].add(p);
      }
      
      private void dfsEuler()
      {
          timeRange = new int[V][2];          
          t = 1;
          dfs(root);           
      }
      
      private void dfs(int node)
      {
          if (dump)
              System.out.println("dfs on node " + node + ", at time " + t);
          timeRange[node][0] = t;
          
          for (int next : nbs[node]) {
              if (timeRange[next][0] == 0)
              {
                  ++t;
                  dfs(next);                  
              }
          }
          timeRange[node][1] = t;
          subTreeSize[node] = t - timeRange[node][0] + 1;
      }
      
      public List<Integer> getNeighbors(int p) {
          return nbs[p];
      }
      
      public int[] getSubTreeSize(){
          return subTreeSize;
      }
      
      public int[][] getTimeRange()
      {
          if (dump){
              for (int i = 0; i < V; i++){
                  System.out.println(i + ": " + timeRange[i][0] + " - " + timeRange[i][1]);
              }
          }
          return timeRange;
      }
      
  }
  
  /* segment tree */
  class SegTree {
      int[] a;
      int[] tree;
      int[] treeMin;
      int[] treeMax;
      int[] lazy;
      int n;
      boolean dump = false;
      
      public SegTree(int n)
      {
          if (dump)
              System.out.println("create segTree with size " + n);
          this.n = n;
          treeMin = new int[n*4];
          treeMax = new int[n*4];
          lazy = new int[n*4];
      }
      
      public SegTree(int n, int[] a) {
          this(n);
          this.a = a;
          buildTree(1, 0, n-1);          
      }
      
      private void buildTree(int node, int lo, int hi)
      {
          if (lo == hi) {
              tree[node] = lo;
              return;
          }
          
          int m = (lo + hi) / 2;
          buildTree(node * 2, lo, m);
          buildTree(node * 2 + 1, m + 1, hi);
          pushUp(node, lo, hi);
      }
      
      
      private void pushUp(int node, int lo, int hi)
      {
          if (lo >= hi) return;
          
          // note that, if we need to call pushUp on a node, then lazy[node] must be zero.
          
          //the true value is the value + lazy
          treeMin[node] = Math.min(treeMin[node * 2] + lazy[node * 2], treeMin[node * 2 + 1] + lazy[node * 2 + 1]);
          treeMax[node] = Math.max(treeMax[node * 2] + lazy[node * 2], treeMax[node * 2 + 1] + lazy[node * 2 + 1]);
          // add combine fcn
      }
      
      private void pushDown(int node, int lo, int hi)
      {
          if (lazy[node] == 0) return;
          
          int lz = lazy[node];
          lazy[node] = 0;          
          
          
          treeMin[node] += lz;
          treeMax[node] += lz;     
          
          if (lo == hi) return;
          
          int mid = (lo + hi) / 2;
          lazy[node * 2] += lz;
          lazy[node * 2 + 1] += lz;
      }
      
      public int rangeQueryMax(int fr, int to)
      {
          return rangeQueryMax(1, 0, n-1, fr, to);
      }
      
      public int rangeQueryMax(int node, int lo, int hi, int fr, int to)
      {
          if (lo == fr && hi == to)
              return treeMax[node] + lazy[node];
          
          int mid = (lo + hi) / 2;
          pushDown(node, lo, hi);
          
          if (to <= mid) {
              return rangeQueryMax(node * 2, lo, mid, fr, to);
          }else if (fr > mid)
              return rangeQueryMax(node * 2 + 1, mid + 1, hi, fr, to);
          else {
              return Math.max(rangeQueryMax(node * 2, lo, mid, fr, mid),
                              rangeQueryMax(node * 2 + 1, mid + 1, hi, mid + 1, to));              
          }
      }
      
      public int rangeQueryMin(int fr, int to)
      {
          return rangeQueryMin(1, 0, n-1, fr, to);
      }
      
      public int rangeQueryMin(int node, int lo, int hi, int fr, int to)
      {
          if (lo == fr && hi == to)
              return treeMin[node] + lazy[node];
          
          int mid = (lo + hi) / 2;
          pushDown(node, lo, hi);
          
          if (to <= mid) {
              return rangeQueryMin(node * 2, lo, mid, fr, to);
          }else if (fr > mid)
              return rangeQueryMin(node * 2 + 1, mid + 1, hi, fr, to);
          else {
              return Math.min(rangeQueryMin(node * 2, lo, mid, fr, mid),
                              rangeQueryMin(node * 2 + 1, mid + 1, hi, mid + 1, to));              
          }
      }
      
       public void rangeUpdate(int fr, int to, int delta){
           rangeUpdate(1, 0, n-1, fr, to, delta);
       }
       
       
      public void rangeUpdate(int node, int lo, int hi, int fr, int to, int delta){
          pushDown(node, lo, hi);
          if (fr == lo && to == hi)
          {
              lazy[node] = delta;
              return;
          }
          
          int m = (lo + hi) / 2;
          if (to <= m) 
              rangeUpdate(node * 2, lo, m, fr, to, delta);
          else if (fr > m)
              rangeUpdate(node * 2 + 1, m + 1, hi, fr, to, delta);
          else  {
              rangeUpdate(node *  2, lo, m, fr, m, delta);
              rangeUpdate(node * 2 + 1, m + 1, hi, m + 1, to, delta);              
          }
          
          // re-set the in-variant
          pushUp(node, lo, hi);
      }
      
      
      
      public int query(int node, int lo, int hi, int fr, int to)
      {
          if (fr == lo && to == hi)
              return tree[node];
          
          int m = (lo + hi) / 2;
          if (to <= m) 
              return  query(node * 2, lo, m, fr, to);
          else if (fr > m)
              return query(node * 2 + 1, m + 1, hi, fr, to);
          
          int lid = query(node * 2, lo, m, fr, m);
          int rid = query(node * 2 + 1, m + 1, hi, m + 1, to);
          return a[lid] >= a[rid] ? lid : rid;
      }
      
      
  }
  
  
  private long inv(long v)
  {
      return pow(v, MOD-2);
  }
  
  private long pow(long v, int p)
  {
      long ans = 1;
      while (p > 0)
      {
          if (p % 2 == 1)
              ans = ans * v % MOD;
          v = v * v % MOD;
          p = p / 2;
      }
      return ans;
  }
  
  private double dist(double x, double y, double xx, double yy)
  {
      return Math.sqrt((xx - x) * (xx - x) + (yy - y) * (yy - y));
  }
  

  
 
  private int mod_add(int a, int b) {
      int v = a + b;
      if (v >= MOD) v -= MOD;
      return v;
  }
  
  
  private long overlap(int x1, int y1, int x2, int y2, int x3, int y3, int x4, int y4) {
      if (x1 > x3) return overlap(x3, y3, x4, y4, x1, y1, x2, y2);
      
      if (x3 > x2 || y4 < y1 || y3 > y2) return 0L;
      
      //(x3, ?, x2, ?)
      int yL = Math.max(y1, y3);
      int yH = Math.min(y2, y4); 
      int xH = Math.min(x2, x4);
      
      return f(x3, yL, xH, yH);
  }
  //return #black cells in rectangle
  private long f(int x1, int y1, int x2, int y2) {
      long dx = 1L + x2 - x1;
      long dy = 1L + y2 - y1;
      if (dx % 2 == 0 || dy % 2 == 0 || (x1 + y1) % 2 == 0) 
          return 1L * dx * dy / 2;
      return 1L * dx * dy / 2 + 1;
  }
  
  private int distM(int x, int y, int xx, int yy) {
      return abs(x - xx) + abs(y - yy);
  }
  
  private boolean less(int x, int y, int xx, int yy) {
      return x < xx || y > yy;
  }
  
  private int mul(int x, int y) {
      return (int)(1L * x * y % MOD);
  }
  
  
 
  
  private int nBit1(int v) {
      int v0 = v;
      int c = 0;
      while (v != 0) {
          ++c;
          v = v & (v - 1);
      }
      return c;
  }
  
  private long abs(long v) {
      return v > 0 ? v : -v;
  }
  
  private int abs(int v) {
      return v > 0 ? v : -v;
  }
  
  private int common(int v) {
      int c = 0;
      while (v != 1) {
          v = (v >>> 1);
          ++c;
      }
      
      return c;
  }
  
  private void reverse(char[] a, int i, int j) {
      while (i < j) {
          swap(a, i++, j--);
      }
  }
  
  private void swap(char[] a, int i, int j) {
      char t = a[i];
      a[i] = a[j];
      a[j] = t;
  }
  
  private int gcd(int x, int y) {
      if (y == 0) return x;
      return gcd(y, x % y);
  }
  private long gcd(long x, long y) {
      if (y == 0) return x;
      return gcd(y, x % y);
  }
  private int max(int a, int b) {
      return a >  b ? a : b;
  }
  
  private long max(long a, long b) {
      return a >  b ? a : b;
  }
  
  private int min(int a, int b) {
      return a >  b ? b : a;
  }
  
  private long min(long a, long b) {
      return a >  b ? b : a;
  }
  
  static class FastReader
    {
        BufferedReader br;
        StringTokenizer st;
 
        public FastReader(BufferedReader in)
        {            
            br = in;
        }
 
        String next()
        {
            while (st == null || !st.hasMoreElements())
            {
                try
                {
                    String line = br.readLine();
                    if (line == null || line.length() == 0) return "";
                    st = new StringTokenizer(line);
                }
                catch (IOException  e)
                {
                    return "";
                    //e.printStackTrace();
                }
            }
            return st.nextToken();
        }
 
        int nextInt()
        {
            return Integer.parseInt(next());
        }
 
        long nextLong()
        {
            return Long.parseLong(next());
        }
 
        double nextDouble()
        {
            return Double.parseDouble(next());
        }
 
        String nextLine()
        {
            String str = "";
            try
            {
                str = br.readLine();
            }
            catch (IOException e)
            {
                return null;
                //e.printStackTrace();
            }
            return str;
        }
    }
}