
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 a circle $C$ that passes through points $A(-2,0)$ and $B(0,2)$, with its center $C$ located on the line $y=x$. Additionally, the line $l$: $y=kx+1$ intersects with circle $C$ at points $P$ and $Q$.

1. Find the equation of circle $C$.
2. If $\overrightarrow{OP} \cdot \overrightarrow{OQ} = -2$, find the value of the real number $k$.
3. Draw a dynamic line $m$ through the point $(0,4)$, intersecting with circle $C$ at points $E$ and $F$. Determine whether there exists a circle $P$ among all circles with $EF$ as the diameter that passes through point $M(2,0)$. If it exists, find the equation of circle $P$; if not, explain the reason.

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