
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
In the plane rectangular coordinate system $xOy$, given the equation of the moving circle $M$ as $\left(x+a+1\right)^{2}+\left(y-2a+1\right)^{2}=1\left(a\in R\right)$, then the equation of the locus of the center of circle $M$ is _____. If for any point $P$ on circle $M$, there exists a point $Q$ on circle $O:x^{2}+y^{2}=4$ such that $\angle OPQ=30^{\circ}$, then the length of the locus of the centers of circle $M$ that satisfy the condition is _____.

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