
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, define points $A = (0, 0)$, $B = (8, 0)$, and $C = (20, 0)$. Line $\ell_A$ has a slope of 2 and passes through $A$. Line $\ell_B$ is vertical and passes through $B$. Line $\ell_C$ has slope $-2$ and passes through $C$. The lines $\ell_A$, $\ell_B$, and $\ell_C$ start rotating clockwise about their respective points $A$, $B$, and $C$. Line $\ell_A$ and $\ell_C$ rotate at the same angular rate, while line $\ell_B$ rotates at half this rate. Determine the largest possible area of the triangle formed by these three lines at any time during their rotation.

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