
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
Ladder of twos. Fixing \( n \), consider a "ladder" of \( n \) twos:

\[
\underbrace{2 + 2 + 2 + \ldots + 2}_{n \text{ times}}
\]

Denote by \( N_n \) the number of distinct integers that can be obtained from this ladder by appropriately and unambiguously placing nested parentheses. For example, \( N_3 = 1 \), \( N_4 = 2 \). Find \( N_n \).

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