
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 hyperbola $C$ passes through the points $P(3,4)$ and $Q(\sqrt{2},\sqrt{2})$.  
$(1)$ Find the standard equation of the hyperbola $C$.  
$(2)$ Given $A(2,-2)$, a line $l$ passing through the point $(2,0)$ intersects the hyperbola $C$ at two distinct points $M$ and $N$. If a circle with diameter $MN$ passes through point $A$, find the equation of the line $l$.

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