
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
A polynomial \( f(x)=\sum_{i=0}^{n} a_{i} x^{i} \) of degree \( n \) or less is called happy if:

(i) Each coefficient \( a_{i} \) satisfies \( 0 \leq a_{i} < 1 \);

(ii) \( f(x) \) is an integer for all integers \( x \).

Find the number of happy polynomials of degree \( n \) or less.

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