
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 $\sqrt{3}$ and passing through the point $P(2,2)$.<br/>$(1)$ Find the standard equation of the hyperbola $C$;<br/>$(2)$ A line $l$ passing through the point $\left(0,4\right)$ intersects the hyperbola $C$ at points $A$ and $B$ (distinct from point $P$), and the lines $PA$ and $PB$ are symmetric with respect to the line $x=2$. Find the distance from point $P$ to the line $l$.

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