
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 $l\_1$: $y=2x+1$.

1. Find the equation of the circle $C$.
2. A line $l$ is drawn through the point $(-1,0)$ and intersects circle $C$ at points $A$ and $B$. $O$ is the coordinate origin. Let $\overrightarrow{OS} = \overrightarrow{OA} + \overrightarrow{OB}$. Determine if there exists a line $l$ such that the diagonals of quadrilateral $OASB$ are equal. If it exists, find the equation(s) of all such line(s) $l$; if not, explain the reason.

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