
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 $C$: $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$) has an eccentricity of $\sqrt {3}$ and the length of its real axis is $2$.

$(1)$ Find the equation of the hyperbola $C$;  
$(2)$ If the line $y=x+m$ intersects the hyperbola $C$ and the length of the chord is $4 \sqrt {2}$, find the value of $m$.

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