
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
There exists a constant $c$, so that among all chords $\overline{AB}$ of the circle $(x-1)^2 + (y-1)^2 = 4$ passing through $C = (0,c),$
\[ t = \sqrt{d_A^2 + d_B^2} \]
where $d_A$ and $d_B$ are the distances from $C$ to $A$ and $B$, respectively, is a fixed constant. Find the constant $t$.

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