
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 $C_{1}: (x+2)^{2}+(y-3)^{2}=1$, and circle $C_{2}: (x-3)^{2}+(y-4)^{2}=9$, where $A$ and $B$ are moving points on circles $C_{1}$ and $C_{2}$ respectively, and point $P$ is a moving point on the $y$-axis, determine the maximum value of $|PB|-|PA|$.

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