
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
Given circle $C$: $x^{2}+y^{2}-4x-2y-20=0$, and line $l$: $4x-3y+15=0$. The line intersects the circle $C$ at points $A$ and $B$. $D$ is any point on the circle $C$ that is different from points $A$ and $B$. Determine the maximum area of triangle $ABD$.

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