
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 circular table are sitting  $ 2n$  people, equally spaced in between.  $ m$  cookies are given to these people, and they give cookies to their neighbors according to the following rule.

(i) One may give cookies only to people adjacent to himself.
(ii) In order to give a cookie to one's neighbor, one must eat a cookie.

Select arbitrarily a person  $ A$  sitting on the table. Find the minimum value  $ m$  such that there is a strategy in which  $ A$  can eventually receive a cookie, independent of the distribution of cookies at the beginning.

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