
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 $C\_1$: $x^2+y^2+6x=0$ is symmetric about line $l\_1$: $y=2x+1$, find the equation of the circle $C$.
(1) Find the equation of circle $C$;
(2) A line $l$ passing through point $(-1,0)$ intersects circle $C$ at points $A$ and $B$. $O$ is the coordinate origin. Let $\overset{→}{OS}= \overset{→}{OA}+ \overset{→}{OB}$, determine if there exists such a line $l$ that makes the diagonals of quadrilateral $OASB$ equal. If it exists, find the equation of all lines $l$ that satisfy the condition; if not, explain the reason.

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