
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
Let $ABCD$ be a trapezoid with $\overline{AD}||\overline{BC}$ and $\angle ABC = \dfrac{\pi}{4}$. The diagonals of the trapezoid are each $20\sqrt{2}$. If point $P$ is on $\overline{AD}$ such that distances from $A$ to $P$ and from $D$ to $P$ are $8\sqrt{2}$ and $16\sqrt{2}$ respectively, find the length of the line segment $PB$.

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