
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 system $xOy$, there is a given circle $C: x^2 + (y-3)^2 = 2$. Point $A$ is a moving point on the $x$-axis, and $AP$ and $AQ$ are tangents to the circle $C$ at points $P$ and $Q$, respectively. Then, the range of the length of line segment $PQ$ is __________.

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