
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 natural number. We call a sequence \$a_1, a_2, \ldots, a_n\$ interesting if for each \$i = 1, 2, \ldots, n\$ one of the following equations holds: \$a_i = i\$ or \$a_i = i + 1\$. We call an interesting sequence even if the sum of its elements is even, and odd otherwise. For each odd interesting sequence, we find the product of its elements and write it on the first sheet. For each even interesting sequence, we do the same and write it on the second sheet. On which sheet is the sum of the numbers greater and by how much? (Give your answer depending on \$n\$.)

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