
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 that point $P$ is the intersection point of the lines $l_{1}$: $mx-ny-5m+n=0$ and $l_{2}$: $nx+my-5m-n=0$ ($m$,$n\in R$, $m^{2}+n^{2}\neq 0$), and point $Q$ is a moving point on the circle $C: \left(x+1\right)^{2}+y^{2}=1$, find the maximum value of $|PQ|$.

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