
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 $AP : PB = 1 : 4$. Point $Q$ lies on side $BC$ of the square and divides it in the ratio $BQ : QC = 5$. Lines $DP$ and $AQ$ intersect at point $E$. Find the ratio of the lengths $AE : EQ$.

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