
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 curve $C$ on the right side of the $y$-axis, the difference between the distance from each point on $C$ to the point $F(1,0)$ and its distance to the $y$-axis is always $1$.

$(1)$ Find the equation of curve $C$.

$(2)$ Does there exist a positive number $m$, such that for any line passing through point $M(m,0)$ and intersecting curve $C$ at two points $A$ and $B$, the dot product $\overrightarrow{FA} \cdot \overrightarrow{FB} < 0$? If it exists, find the range of $m$; if not, explain why.

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