
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 the hyperbola $C$: $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt 0,b \gt 0\right)$ with eccentricity $\sqrt{3}$ and length of the real axis as $2$.  
$(1)$ Find the equation of the hyperbola $C$;  
$(2)$ If the chord cut by the line $y=x+m$ on the hyperbola $C$ has a length of $4\sqrt{2}$, find the value of $m$.

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