
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 $F(1,0)$, the line $l:x=-1$, and $P$ as a moving point on the plane. A perpendicular line is drawn from point $P$ to line $l$, with the foot of the perpendicular being point $Q$, and it is known that $\overrightarrow{QP} \cdot \overrightarrow{QF} = \overrightarrow{FP} \cdot \overrightarrow{FQ}$.

$(1)$ Find the equation of the trajectory curve $C$ of the moving point $P$;

$(2)$ Suppose a moving line $y=kx+m$ is tangent to curve $C$ at point $M$, and intersects with the line $x=-1$ at point $N$. Investigate whether there exists a fixed point $E$ in the coordinate plane such that the circle with diameter $MN$ always passes through this fixed point $E$. If it exists, find the coordinates of the fixed point $E$; if not, explain why.

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