
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
Point $O$ lies on plane $\alpha$, and $A$, $B$, and $C$ are three non-collinear points on plane $\alpha$. The moving point $P$ on plane $\alpha$ satisfies $\overrightarrow{OP} = \overrightarrow{OA} + \lambda (\overrightarrow{AB} + \overrightarrow{AC})$. When the value of $\lambda$ makes $\overrightarrow{PA} \cdot (\overrightarrow{PB} + \overrightarrow{PC}) = 0$, determine the value of $\lambda$.

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