
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 $C:\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1(a > 0,b > 0)$ with asymptote equations $y=\pm \sqrt{3}x$ and right vertex at $\left( 1,0 \right)$.

(I) Find the equation of hyperbola $C$;

(II) Given line $y=x+m$ intersects hyperbola $C$ at two distinct points $A,B$, and the midpoint of line segment $AB$ is $M\left( {{x}\_{0}},{{y}\_{0}} \right)$, find the value of $\frac{{{y}\_{0}}}{{{x}\_{0}}}$ when ${{x}\_{0}}\neq 0$.

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