
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

Extend line $PD$ to intersect line $BC$ at point $F$. Construct lines $DG$ and $PT$ parallel to $AQ$.

Introduce the following notations: $AP = x$, $PB = \lambda x$, $BQ = y$, $QC = \mu y$, $PE = u$, $ED = v$.

From the similarity of triangles:

\[
\frac{z}{(1+\mu)y} = \lambda, \quad z = \lambda(1+\mu)y
\]
\[
\frac{BT}{\lambda x} = \frac{y}{(1+\mu)y}, \quad BT = \frac{\lambda x}{1+\mu} = \frac{\lambda y}{1+\lambda}, \quad TQ = y - \frac{\lambda y}{1+\lambda} = \frac{y}{1+\lambda}
\]
\[
QG = AD = (1+\mu)y
\]

By Thales' theorem:

\[
\frac{PE}{ED} = \frac{u}{v} = \frac{TQ}{QG} = \frac{y}{(1+\lambda)(1+\mu)y} = \frac{1}{(1+\lambda)(1+\mu)} = 3:20
\]

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