
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 \( C_1 \): \( (x-1)^2+(y+1)^2=1 \), and circle \( C_2 \): \( (x-4)^2+(y-5)^2=9 \), find the maximum value of \( |PN|-|PM| \), where \( P \) is a point on the \( x \)-axis, and \( M \) and \( N \) are points on circles \( C_1 \) and \( C_2 \) respectively.

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