
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 a triangle $\mathrm{PEF}$ with sides $\mathrm{PE}=3$, $\mathrm{PF}=5$, and $\mathrm{EF}=7$. A segment $\mathrm{PA}=1.5$ is laid on the extension of the side $\mathrm{FP}$ beyond point $\mathrm{P}$. Find the distance $d$ between the centers of the circumcircles of triangles $\mathrm{EPA}$ and $\mathrm{EAF}$. Provide the number equal to $2d$.

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