
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
Let  $ \triangle ABE $  be a triangle with  $ \frac{AB}{3} = \frac{BE}{4} = \frac{EA}{5} $ . Let  $ D \neq A $  be on line  $ \overline{AE} $  such that  $ AE = ED $  and  $ D $  is closer to  $ E $  than to  $ A $ . Moreover, let  $ C $  be a point such that  $ BCDE $  is a parallelogram. Furthermore, let  $ M $  be on line  $ \overline{CD} $  such that  $ \overline{AM} $  bisects  $ \angle BAE $ , and let  $ P $  be the intersection of  $ \overline{AM} $  and  $ \overline{BE} $ . Compute the ratio of  $ PM $  to the perimeter of  $ \triangle ABE $ .

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