
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 Cartesian coordinate system with origin $O$, point $A(4, -3)$ is the right-angle vertex of $\triangle OAB$, given that $|AB| = 2|OA|$, and the ordinate of point $B$ is greater than $0$.

$(1)$ Find the coordinates of $\overrightarrow{AB}$;

$(2)$ Find the equation of the circle symmetric to the circle $x^{2} - 6x + y^{2} + 2y = 0$ with respect to the line $OB$.

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