
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
In a Cartesian coordinate system, point P is a moving point on the line $l: x = -1$, point F is at $(1,0)$, and point Q is the midpoint of segment PF. Point M is such that $MQ \perp PF$ and $\overrightarrow{MP} = \lambda \overrightarrow{OF}$. A circle centered at $(3,0)$ with an equation $(x-3)^2 + y^2 = 2$ has tangents through point M with tangent points A and B. The minimum value of $|AB|$ is (　).

A) 3  
B) $\frac{3}{2}$  
C) $\frac{\sqrt{6}}{2}$  
D) $\sqrt{6}$

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