
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 $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ ($a>0, b>0$) with a real axis length of $2$ and a right focus at $({\sqrt{5},0})$.  
$(1)$ Find the equation of the hyperbola $C$.  
$(2)$ Given that the line $y=x+2$ intersects the hyperbola $C$ at two distinct points $A$ and $B$, find $|AB|$.

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