
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 the circle $C$: $(x-1)^2 + (y-2)^2 = 2$, and the point $P$ with coordinates $(2, -1)$. A tangent to circle $C$ passes through point P, and the points of tangency are $A$ and $B$.
(1) Find the equations of the tangent lines $PA$ and $PB$.
(2) Determine the length of the tangent from point P to the circle.
(3) Find the equation of line $AB$.

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