
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 $ABCDEFGH$, and $Y$ is a point on the side $\overline{AB}$ such that $AY:YB = 1:3$. What fraction of the area of the octagon is shaded when triangles $\triangle BCO$, $\triangle CDO$, and half of $\triangle ABO$ are shaded?
A) $\frac{1}{4}$
B) $\frac{7}{32}$
C) $\frac{9}{32}$
D) $\frac{10}{32}$
E) $\frac{11}{32}$

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