
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 a line $l$ intersects the parabola $y^2=4x$ at two distinct points $A$ and $B$, and is tangent to the circle $(x-5)^2+y^2=r^2$ ($r>0$) at point $M$, and $M$ is the midpoint of segment $AB$.  
(1) If $\triangle AOB$ is an equilateral triangle ($O$ is the origin of coordinates), find the side length of this triangle;  
(2) If $r=4$, find the equation of line $l$;  
(3) Discuss for $r\in(0, +\infty)$, please write down the number of lines $l$ that meet the conditions (just write the result directly).

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