
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 the first day, $2^n$ students played a knockout table tennis tournament: first, two played, then the winner played against the third, the winner of that pair played against the fourth, and so on, until the last student played (draws are not possible in table tennis).

On the second day, the same students competed for a cup: they randomly paired up and played in pairs, the losers were eliminated, and the winners were randomly paired up again and played in pairs, and so on. It turned out that the sets of pairs that played on the first and second days were the same (possibly with different winners). Find the largest possible value of $n$.

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