
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
It is known that for any point on curve $C$, the difference between its distance to point $F(0,1)$ and its distance to the $x$-axis is always $1$.

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

$(2)$ Let the line $y=kx+m(m > 0)$ intersect curve $C$ at points $A$ and $B$. If for any $k \in \mathbb{R}$, it holds that $\overrightarrow{FA} \cdot \overrightarrow{FB} < 0$, find the range of $m$.

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