
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 an equilateral triangle $ABC$ with side length $2$, and $P$ is a point on the plane of triangle $ABC$.
- $(1)$ If $\overrightarrow{PC} = -(\overrightarrow{PA} + \overrightarrow{PB})$, find the area of $\triangle PAB$.
- $(2)$ If $\overrightarrow{PB} \cdot \overrightarrow{PC} = 0$, find the maximum value of $|\overrightarrow{PB}| + |\overrightarrow{PC}|$.
- $(3)$ Find the minimum value of $2\overrightarrow{PA} \cdot \overrightarrow{PB} + \overrightarrow{PA} \cdot \overrightarrow{PC}$.

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