
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
We call a rational number "good" if it satisfies the following conditions: \( x = \frac{p}{q} > 1 \), where \( p \) and \( q \) are coprime positive integers, and there exist constants \( \alpha \) and \( N \) such that for any positive integer \( n \geqslant N \), the inequality holds:
$$
\left|\left\{ x^n \right\} - \alpha \right| \leqslant \frac{1}{2(p+q)},
$$
where \( \{a\} \) represents the fractional part of \( a \). Determine all "good" rational numbers.

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