
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
(The full score for this question is 12 points) It is known that line $l$ passes through the focus $F$ of the parabola $y^2=4x$, and intersects the parabola at points $A$ and $B$.
(1) If $|AF|=4$, find the coordinates of point $A$;
(2) Let the slope of line $l$ be $k$. When the length of segment $AB$ equals 5, find the value of $k$.
(3) Find the minimum distance from a point $P$ on the parabola $y^2=4x$ to the line $2x-y+4=0$, and find the coordinates of point $P$ at this time.

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