
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
$ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD$. The crease is $EF$, where $E$ is on $AB$ and $F$ is on $CD$. The dimensions $AE=12, BE=25$, and $CF=5$ are given. Determine the dimensions of the sides and the perimeter of rectangle $ABCD$. You may leave the size as a rational expression simplified in terms of relative prime integers $m/n$ and find $m+n$.

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