
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 ellipse $C_{1}$ and parabola $C_{2}$ have a common focus $F(1,0)$, the center of $C_{1}$ and the vertex of $C_{2}$ are both at the origin. A line $l$ passing through point $M(4,0)$ intersects the parabola $C_{2}$ at points $A$ and $B$ (where point $A$ is in the fourth quadrant).
$(1)$ If $|MB|=4|AM|$, find the equation of line $l$;
$(2)$ If the symmetric point $P$ of the origin $O$ with respect to line $l$ is on the parabola $C_{2}$, and line $l$ has common points with ellipse $C_{1}$, find the minimum length of the major axis of ellipse $C_{1}$.

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