
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 two circles, $\odot O_1: (x-1)^2+y^2=4$ and $\odot O_2: x^2+(y-\sqrt{3})^2=9$.

(1) Find the equation of the line on which the common chord of the two circles lies.  
(2) Find the length of the common chord of the two circles.

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