
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
In the coordinate plane, consider points \( A = (0, 0) \), \( B = (10, 0) \), and \( C = (15, 0) \). Line \( \ell_A \) has slope \( 1 \) and passes through \( A \). Line \( \ell_B \) is vertical and passes through \( B \). Line \( \ell_C \) has slope \( -1 \) and passes through \( C \). All three lines \( \ell_A \), \( \ell_B \), and \( \ell_C \) begin rotating clockwise about points \( A \), \( B \), and \( C \), respectively, at the same angular rate. At any given moment, the three lines form a triangle. What is the largest possible area of such a triangle?

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