
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 $M_{1}$: $\left(x-2\right)^{2}+\left(y-1\right)^{2}=2$, circle $M_{2}$: $x^{2}+y^{2}-2x+2y+1=0$, and point $P$ is a moving point on the $y$-axis, find the minimum value of $|PM_{1}|+|PM_{2}|$.

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