
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
Let \(A\) and \(B\) be endpoints of a semicircular arc of radius \(3\). This arc is divided into nine congruent arcs by eight equally spaced points \(C_1, C_2, \dots, C_8\). All cords of the form \(\overline{AC_i}\) or \(\overline{BC_i}\) are drawn. Find the product of the lengths of these sixteen chords.

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