
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
The apex of a regular pyramid with a square base $ABCD$ of unit side length is $E$. Point $P$ lies on the base edge $AB$ and point $Q$ lies on the lateral edge $EC$ such that $PQ$ is perpendicular to both $AB$ and $EC$. Additionally, we know that $AP : PB = 6 : 1$. What are the lengths of the lateral edges?

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