
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
Given the point \( P \) lies in the plane of the right triangle \( \triangle ABC \) with \( \angle BAC = 90^\circ \), and \( \angle CAP \) is an acute angle. Also given are the conditions:
\[ |\overrightarrow{AP}| = 2, \quad \overrightarrow{AP} \cdot \overrightarrow{AC} = 2, \quad \overrightarrow{AP} \cdot \overrightarrow{AB} = 1. \]

Find the value of \( \tan \angle CAP \) when \( |\overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AP}| \) is minimized.

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