
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 two circles $C_{1}$ and $C_{2}$: $(x+1)^{2}+(y+2)^{2}=4$ are symmetric with respect to the line $x-y+1=0$.  
$(1)$ Find the equation of circle $C_{1}$.  
$(2)$ Let the line passing through point $A(0,3)$ intersect circle $C_{1}$ at points $M$ and $N$, where $O$ is the origin. Find the equation of the line $l$ when $\overline{OM}•\overline{ON}=\frac{7}{5}$.

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