
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 the Cartesian coordinate plane, point $P$ is a moving point on the line $x=-1$, point $F(1,0)$, point $Q$ is the midpoint of $PF$, point $M$ satisfies $MQ \perp PF$ and $\overrightarrow{MP}=\lambda \overrightarrow{OF}$, and the tangent line is drawn through point $M$ on the circle $(x-3)^{2}+y^{2}=2$ with tangent points $A$ and $B$, respectively. Find the minimum value of $|AB|$.

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