
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Point $O$ is the center of the regular octagon $ABCDEFGH$, $X$ is the midpoint of side $\overline{AB}$, and $Y$ is the midpoint of side $\overline{CD}$. Calculate the fraction of the area of the octagon that is shaded if the shaded region includes the full area of triangles $\triangle BCO$, $\triangle CDO$, $\triangle DEO$, $\triangle EFO$, and half the area each of $\triangle ABO$ and $\triangle CDO$.
A) $\frac{3}{8}$
B) $\frac{7}{16}$
C) $\frac{13}{32}$
D) $\frac{1}{2}$
E) $\frac{5}{8}$

Present the answer in LaTex format: \boxed{Your answer}
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