
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 \( P \) is located on side \( AB \) of square \( ABCD \) such that \( AP:PB = 1:2 \). Point \( Q \) lies on side \( BC \) of the square and divides it in the ratio \( BQ:QC = 2 \). Lines \( DP \) and \( AQ \) intersect at point \( E \). Find the ratio of lengths \( PE:ED \).

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