
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
In the plane rectangular coordinate system $xOy$, given the hyperbola $C$: $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt 0,b \gt 0\right)$ with the left vertex $A$ and the right focus $F$; point $P(2,3)$ lies on the hyperbola $C$, and the line $l$ intersects the hyperbola $C$ at points $M$ and $N$, such that when the slope of line $MA$ is $1$, $MF=AF$.  
$(1)$ Find the equation of the hyperbola $C$;  
$(2)$ If $OM\bot ON$, find the distance from $O$ to the line $l$.

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