solution: Labeling our vertices will help a great deal, as will drawing a few radii: [asy]
  pair pA, pB, pC, pD, pE, pF, pO;
  pO = (0, 0);
  pA = pO + dir(-10);
  pB = pO + dir(60);
  pC = pO + dir(130);
  pD = pO + dir(170);
  pE = pO + dir(-160);
  pF = pO + dir(-80);
  draw(pA--pB--pC--pD--pE--pF--pA);
  draw(pA--pO--pC--pO--pE--pO, red);
  draw(circle(pO, 1));
  label("$O$", pO, NE);
  label("$A$", pA, E);
  label("$B$", pB, NE);
  label("$C$", pC, NW);
  label("$D$", pD, W);
  label("$E$", pE, SW);
  label("$F$", pF, S);
  label("$105^\circ$", pF, N * 2);
  label("$110^\circ$", pB, SW * 1.5);
  label("$\alpha$", pD, E);
  [/asy] First of all, we see that $\angle ABC = 110^\circ$ must be half of the major arc ${AEC},$ thus arc ${AEC} = 2 \cdot \angle ABC.$ Then, the minor arc ${AC}$ must be $360^\circ - 2 \cdot \angle ABC = 360^\circ - 2 \cdot 110^\circ = 140^\circ.$
  
  Likewise, the minor arc ${EA}$ must be $360^\circ - 2 \cdot \angle EFA = 360^\circ - 2 \cdot 105^\circ = 150^\circ,$ and the minor arc ${CE}$ is $360^\circ - 2 \alpha.$ Now, arc ${AC},$ ${CE},$ and ${EA}$ must add up to $360^\circ,$ which means that \begin{align*}
  360^\circ &= (360^\circ - 2 \alpha) + 140^\circ + 150^\circ\\
  360^\circ &= 650^\circ - 2\alpha\\
  2\alpha &= 290^\circ\\
  \alpha &= \boxed{145^\circ}.
  \end{align*}
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  A hexagon is inscribed in a circle: [asy]
  pair pA, pB, pC, pD, pE, pF, pO;
  pO = (0, 0);
  pA = pO + dir(-10);
  pB = pO + dir(60);
  pC = pO + dir(130);
  pD = pO + dir(170);
  pE = pO + dir(-160);
  pF = pO + dir(-80);
  draw(pA--pB--pC--pD--pE--pF--pA);
  label("$105^\circ$", pF, N * 2);
  label("$110^\circ$", pB, SW * 1.5);
  label("$\alpha$", pD, E);
  draw(circle(pO, 1));
  [/asy] What is the measure of $\alpha$, in degrees?
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 145^\circ
extra_info.question: A hexagon is inscribed in a circle: [asy]
  pair pA, pB, pC, pD, pE, pF, pO;
  pO = (0, 0);
  pA = pO + dir(-10);
  pB = pO + dir(60);
  pC = pO + dir(130);
  pD = pO + dir(170);
  pE = pO + dir(-160);
  pF = pO + dir(-80);
  draw(pA--pB--pC--pD--pE--pF--pA);
  label("$105^\circ$", pF, N * 2);
  label("$110^\circ$", pB, SW * 1.5);
  label("$\alpha$", pD, E);
  draw(circle(pO, 1));
  [/asy] What is the measure of $\alpha$, in degrees?
extra_info.split: test