
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
In a Cartesian coordinate system, for rectangle ABCD, if line AB is defined by the equation $x - 2y - 2 = 0$ and the diagonal line of the rectangle is relative to the point $(0,1)$, then find the length of the chord intercepted on the circle $(x - 1)^2 + (y - 1)^2 = 9$ by line CD.

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