
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 circle $O: x^2 + y^2 = r^2$ ($r > 0$), which intersects the y-axis at points $M$ and $N$, with $M$ above $N$. If the line $y = 2x + \sqrt{5}$ is tangent to circle $O$,

(1) Find the value of the real number $r$;

(2) If a moving point $P$ satisfies $PM = \sqrt{3}PN$, find the maximum value of the area of $\triangle PMN$;

(3) Suppose two distinct points $A$, $B$ on circle $O$ satisfy that the product of the slopes of lines $MA$, $MB$ is $\frac{\sqrt{3}}{3}$. Investigate whether line $AB$ passes through a fixed point. If it does, find the coordinates of the fixed point; if not, explain why.

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