
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 circle $C_{1}$: $\left(x-2\right)^{2}+\left(y-3\right)^{2}=1$, circle $C_{2}$: $\left(x--3\right)^{2}+\left(y-4\right)^{2}=9$, $M$ and $N$ are moving points on circles $C_{1}$ and $C_{2}$ respectively, $P$ is a moving point on the $x$-axis. Find the maximum value of $|PN|-|PM|$.

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