
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 three points on the coordinate plane: \( A = (0, 0) \), \( B = (17, 0) \), and \( C = (23, 0) \). Line \( \ell_A \) passes through \( A \) with a slope of 2. Line \( \ell_B \) is vertical and passes through \( B \). Line \( \ell_C \) passes through \( C \) with a slope of \(-2\). These lines start rotating clockwise with the same angular velocity around their respective points. Determine the maximum area of any triangle formed by these three lines at any instance during their rotation.

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