
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 that $F$ is the focus of the parabola $C: y^{2}=4x$, point $P$ lies on $C$, and point $Q$ satisfies $\overrightarrow{PQ}=2\overrightarrow{QF}$. The locus of point $Q$ is curve $E$.  
$(1)$ Find the equation of curve $E$.  
$(2)$ A line $l$ passing through point $F$ intersects curve $E$ at points $M$ and $N$ such that $|MN|=4$. Find the equation of line $l$.

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