
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

Around a bird feeder in the same plane as it, a tit and a bullfinch are flying along two circles at the same speed. A rectangular coordinate system is introduced in this plane, in which the feeder (the common center of the circles) is located at the point $O(0, 0)$. The tit is moving clockwise, and the bullfinch is moving counterclockwise. At the initial moment, the tit and the bullfinch are located at the points $M_{0}(\sqrt{3}, 3)$ and $N_{0}(6, -2\sqrt{3})$ respectively. Determine the coordinates of all positions of the bullfinch where the distance between the birds will be the shortest.

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