
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 hyperbola with its foci on the $x$-axis and an eccentricity of $\sqrt{3}$, and the length of its imaginary axis is $2\sqrt{2}$.

$(1)$ Find the equation of the hyperbola $C$;  
$(2)$ Given a line $x-y+m=0$ intersects the hyperbola $C$ at two distinct points $A$ and $B$. If $OA \perp OB$, find the value of $m$. ($O$ is the origin.)

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