
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 circle $C_1$: $(x-1)^2+(y-2)^2=1$  
(1) Find the equation of the tangent line to circle $C_1$ passing through point $P(2,4)$.  
(2) If circle $C_1$ intersects with circle $C_2$: $(x+1)^2+(y-1)^2=4$ at points $A$ and $B$, find the length of segment $AB$.

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