
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
In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, it is given that $A B = B C = 4$ and $A A_{1} = 2$. Point $P$ lies on the plane $A_{1} B C$, and it holds that $\overrightarrow{D P} \cdot \overrightarrow{P B} = 0$. Find the area of the plane region formed by all such points $P$ satisfying the given condition.

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