
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 \( |\overrightarrow{AB}| = 10 \). If a point \( P \) on the plane satisfies \( |\overrightarrow{AP} - t \overrightarrow{AB}| \geq 3 \) for any \( t \in \mathbb{R} \), then the minimum value of \( \overrightarrow{PA} \cdot \overrightarrow{PB} \) is \(\qquad\), and at this point, \( |\overrightarrow{PA} + \overrightarrow{PB}| = \) \(\qquad\).

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