
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

Examine the accuracy of the following approximation for the side of a regular 9-gon inscribed in circle $k$.

Consider a regular hexagon inscribed in $k$, with vertices $A, B, C, D$ placed consecutively. Let $G$ be the intersection of the circles centered at $B$ and $D$ with radius $BA$ and $DB$ respectively, located inside $k$. Let $H$ be the intersection point of line $BG$ and $AC$. Then, the segment $BH$ approximates the side length of the 9-gon.

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