
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 point $P(2,0)$ and circle $C$: $x^{2}+y^{2}-6x+4y+4=0$.

$(1)$ Suppose line $l_{1}$ passes through $P$ and intersects circle $C$ at points $M$ and $N$. When $|MN|=4$, find the equation of circle $Q$ with diameter $MN$;

$(2)$ Suppose line $ax-y+1=0$ intersects circle $C$ at points $A$ and $B$. Is there a real number $a$ such that the line $l_{2}$ passing through point $P(2,0)$ perpendicularly bisects chord $AB$? If it exists, find the value of $a$; if not, explain why.

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