
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

Set \( A \) in the \( Oxy \) plane is defined by the equation \( x^2 + y^2 = 2x + 2y + 23 \). Set \( B \) in the same plane is defined by the equation \( |x - 1| + |y - 1| = 5 \). Set \( C \) is the intersection of sets \( A \) and \( B \). What is the maximum value of the product of the lengths of \( n \) segments \( XY_1 \cdot XY_2 \cdot XY_3 \cdot \ldots \cdot XY_n \), where point \( X \) is an arbitrarily chosen point from set \( A \), and points \( Y_1, Y_2, Y_3, \ldots, Y_n \) are all elements of set \( C \)?

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