
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 $F_{1}(-2,0)$, $F_{2}(2,0)$, a point $P$ satisfies $|PF_{1}|-|PF_{2}|=2$. Let the locus of point $P$ be denoted as $\Gamma$. A line $l$ with slope $k$ passes through point $F_{2}$ and intersects the locus $\Gamma$ at points $A$ and $B$.

1. Find the equation of the locus $\Gamma$.
2. Determine the range of values for the slope $k$.
3. Does there exist a fixed point $M$ on the $x$-axis such that for any line $l$ passing through $F_{2}$, $MA\bot MB$ always holds true? If it exists, find the point $M$; if not, explain why.

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