
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
Consider a circle of radius 3, divided into 8 equal arcs by the points $P_1, P_2, \dots, P_7$. Let $A$ and $B$ be the endpoints of the semicircular arc (diameter). You draw all chords of the form $\overline{AP_i}$ and $\overline{BP_i}$. Determine the product of the lengths of these fourteen chords.

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