
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
In the given diagram, let \( E \) and \( F \) be the midpoints of sides \( AD \) and \( BC \) of rectangle \( ABCD \) respectively. Points \( M \) and \( N \) are on \( AB \) and \( CF \) respectively, with \( MN \) intersecting \( AF \) at point \( P \). Given that \( MN = 12 \), \( AD = 2AB \), and \( PE \perp MN \), find the length of \( PE \).

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