
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 equation $x^2+y^2-2x+2my+m^2-2m-2=0$ ($m \in \mathbb{R}$).

(1) If the equation represents a circle, find the range of the real number $m$.

(2) If the circle $C$ represented by the equation has its center at $C(1,1)$, find the equation of the tangent line to circle $C$ that passes through point $P(2,4)$.

(3) If the line $x+y+t=0$ intersects circle $C$ from part (2) at points $A$ and $B$, and $\triangle ABC$ is a right-angled triangle, find the value of the real number $t$.

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