
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 the parabola $C: y^{2}=2px\left(p \gt 0\right)$ with focus $F$, where $A$ is a moving point on $C$, and point $P(1,1)$ is not on $C$, and $|AF|+|AP|$ has a minimum value of $2$.<br/>$(1)$ Find the equation of $C$;<br/>$(2)$ If the line $AP$ intersects $C$ at another point $B$ and intersects the line $l$ at point $Q$, where $\overrightarrow{QA}=\lambda \overrightarrow{PA}$, $\overrightarrow{QB}=\mu \overrightarrow{PB}$, and $\lambda +\mu =4$, find the equation of the line $l$.

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