
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 a circle $C: x^2 + y^2 + x - 6y + m = 0$,

$(1)$ When $m=\frac{1}{4}$, a ray of light $L$ emitted from point $A\left(\frac{11}{2},3\right)$ hits the $x$-axis and is reflected. The line of the reflected ray is tangent to circle $C$. Find the equation of the line of the reflected ray.

$(2)$ The line $x + 2y - 3 = 0$ intersects circle $C$ at points $P$ and $Q$. If $\vec{OP} \cdot \vec{OQ} = 0$ (where $O$ is the origin), find the value of $m$.

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