
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 point F (-2, 0) is inside a circle O with the origin as its center, and the shortest chord through F has a length of 2.
(1) Find the equation of circle O;
(2) For any chord AB through F that is not perpendicular to either coordinate axis, if point M is on the x-axis, and MF bisects the interior angle at M of triangle AMB, find the coordinates of point M.

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