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

$(1)$ Find the coordinates of vector $\vec{AB}$.

$(2)$ Find the equation of the circle which is symmetric to the circle $x^2 - 6x + y^2 + 2y = 0$ about the line $OB$.

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