
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 $A = (0, 0)$, $B = (7, 0)$, $C = (15, 0)$, and $D = (20,0)$. 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 a slope of -2 and passes through $C$. Line $\ell_D$ has a slope of 2 and passes through $D$. All these lines begin rotating clockwise about their respective points at the same angular rate. The lines $\ell_A$, $\ell_B$, $\ell_C$, and $\ell_D$ create various triangles as they rotate. Determine the largest possible area of triangle $XYZ$, where $X = \ell_A \cap \ell_B$, $Y = \ell_C \cap \ell_D$, and $Z = \ell_B \cap \ell_C$.

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