
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 that \( P \) is a moving point on the circle \((x-2)^2 + (y-2)^2 = 1\), the vector \(\overrightarrow{OP}\) is rotated counterclockwise by \(90^\circ\) to obtain the vector \(\overrightarrow{OS}\). Additionally, the point \( P \) is reflected about \(A(3,0)\) to obtain point \( T \). Find the range of the length \(|\overrightarrow{TS}|\).

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