
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
Point $E$ is on line segment $\overline{AD}$ such that $AE = 15$ and $ED = 25$. Point $F$ is not on $\overline{AD}$ such that $AF = FD$, and both $AF$ and $EF$ are integers. Additionally, $AF^2$ is a perfect square. Find the sum $s$ of all possible perimeters of $\triangle AFD$.

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