
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 $C$ passes through the point $(1,\sqrt{3})$, the center of the circle is on the line $y=x$, and the chord intercepted by the line $y=-x+2$ has a length of $2\sqrt{2}$.

$(1)$ Find the equation of circle $C$:

$(2)$ If line $l$ passes through the point $\left(\dfrac{3}{2},0\right)$, and intersects circle $C$ at points $P$ and $Q$, and $\overrightarrow{OP} \cdot \overrightarrow{OQ}=-2$, find the equation of line $l$.

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