
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
On a table, there is a piece of sugar around which an ant and a beetle crawl along two circles at the same speed. A rectangular coordinate system is introduced on the table's surface, with the sugar (the common center of the circles) at point $O(0 ; 0)$. The ant moves clockwise, while the beetle moves counterclockwise. At the initial moment in time, the ant and the beetle are at points $M_{0}(-1 ; \sqrt{3})$ and $N_{0}(2 \sqrt{3} ; 2)$, respectively. Determine the coordinates of all beetle positions where the distance between it and the ant is the shortest.

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