
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 $m\in R$, let point $P(x,y)$ be the intersection of the family of lines $x+my=0$ passing through point $A$ and the family of lines $mx-y-m+3=0$ passing through point $B$. Find the maximum value of $| \overrightarrow{PA}| \cdot |\overrightarrow{PB}|$.

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