
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 a circle $M$ passes through the point $P(1, -\sqrt{3})$ and is symmetric to the circle $C$: $(x+2)^{2}+y^{2}=r^{2} (r > 0)$ with respect to the $y$-axis.

$(I)$ Find the equation of circle $M$;  
$(II)$ If there are two mutually perpendicular lines $l_{1}$ and $l_{2}$, both passing through the point $A(-1,0)$, and the lengths of the chords intercepted by circle $C$ on $l_{1}$ and $l_{2}$ are $d_{1}$ and $d_{2}$ respectively, find the value of $d_{1}^{2}+d_{2}^{2}$.

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