
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 circles $C_1: (x+6)^2+(y-5)^2=4$ and $C_2: (x-2)^2+(y-1)^2=1$, let $M$ and $N$ be moving points on circles $C_1$ and $C_2$ respectively, and let $P$ be a moving point on the $x$-axis. Find the minimum value of $|PM|+|PN|$.

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