
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 $C: (x-2)^2 + y^2 = 1$, point $P$ is a moving point on the line $l: x+y=0$. Two tangents are drawn from point $P$ to circle $C$, with points of tangency at $A$ and $B$.  
$(1)$ If the coordinates of $P$ are $P(-1,1)$, find the equation of the tangent passing through point $P$.  
$(2)$ Determine whether the line $AB$ always passes through a fixed point. If so, find this fixed point; if not, explain the reason.  
$(3)$ The line $x-y+m=0$ intersects circle $C$ at points $E$ and $F$. Find the range of values of $\overrightarrow{OE} \cdot \overrightarrow{OF}$ (where $O$ is the origin).

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