
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 eccentricity $2$ and passing through the point $A\left(2,3\right)$.  
$(1)$ Find the equation of the hyperbola $C$;  
$(2)$ If the line $l$ with slope $\frac{\sqrt{5}}{5}$ intersects $C$ at points $P$ and $Q$, and intersects the $x$-axis at point $M$, where $Q$ is the midpoint of $PM$, find the equation of $l$.

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