
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
Let \( n \) be a nonzero natural number. How many sequences \( a_{1}, a_{2}, \ldots, a_{n^{2}+n} \) of integers equal to 0 or 1 are there such that

$$
a_{i+1}+a_{i+2}+\cdots+a_{i+n}<a_{i+n+1}+a_{i+n+2}+\cdots+a_{i+2n}
$$

for every integer \( i \) such that \( 0 \leq i \leq n^{2} - n \)?

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