
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
Find the smallest positive integer \( n \) such that the polynomial \( (x+1)^{n} - 1 \) is divisible by \( x^2 + 1 \) modulo 3. 

This can be interpreted through either of the following equivalent conditions:
- There exist polynomials \( P(x) \) and \( Q(x) \) with integer coefficients such that \( (x+1)^{n} - 1 = (x^2 + 1) P(x) + 3 Q(x) \).
- The remainder when \( (x+1)^{n} - 1 \) is divided by \( x^2 + 1 \) is a polynomial with integer coefficients all divisible by 3.

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