
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 $C:y^{2}=4x$, draw two mutually perpendicular lines through its focus $F$ that are not parallel to the coordinate axes. These lines intersect the parabola $C$ at points $P_{1}$, $P_{2}$ and $P_{3}$, $P_{4}$, respectively. The midpoints of line segments $P_{1}P_{2}$ and $P_{3}P_{4}$ are denoted as $M_{1}$ and $M_{2}$, respectively.

$(1)$ Find the equation of the trajectory of the midpoint $M_{1}$ of line segment $P_{1}P_{2}$.

$(2)$ Find the minimum value of the area of $\triangle FM_{1}M_{2}$.

$(3)$ Does the line $l$ passing through $M_{1}$ and $M_{2}$ always pass through a fixed point? If so, find the coordinates of the fixed point; if not, please explain why.

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