
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 that the real axis length of hyperbola $C$ is $4$, and it shares a common focus with the hyperbola $\frac{x^2}{2}-\frac{y^2}{3}=1$.  
$(1)$ Find the equation of hyperbola $C$;  
$(2)$ Given $M(5,0)$, $P$ is an arbitrary point on hyperbola $C$, find the minimum value of $|PM|$.

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