
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 the endpoints of a semicircular arc of radius \(3\). This arc is divided into nine congruent arcs by eight equally spaced points \(D_1, D_2, \ldots, D_8\). Calculate the product of the lengths of all the chords formed by \(\overline{AD_i}\) and \(\overline{BD_i}\) for \(i = 1\) to \(8\).

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