
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
Given a decimal number \( x = a_{1} a_{2} \cdots a_{n} \), where \( n \) is the number of digits, \( a_{1}, a_{2}, \cdots, a_{n} \) are non-negative integers not exceeding 9, and \( a_{1} \neq 0 \). Consider the following sequence of numbers obtained by cyclically permuting \( x \):

\[ x_{1} = x = a_{1} a_{2} \cdots a_{n} \]
\[ x_{2} = a_{n} a_{1} a_{2} \cdots a_{n-1} \]
\[ x_{3} = a_{n-1} a_{n} a_{1} a_{2} \cdots a_{n-2} \]
\[
\cdots \cdots
\]
\[ x_{n} = a_{2} a_{3} \cdots a_{n} a_{1} \]

(For example, if \( n = 5 \) and \( x = 37001 \), then \( x_{1} = 37001, x_{2} = 13700, x_{3} = 01370 (= 1370), x_{4} = 00137 (= 137), x_{5} = 70013 \)).

Determine the smallest number \( n \) such that \( x = a_{1} a_{2} \cdots a_{n} \) yields \( n \) numbers that are all divisible by 1989 after the cyclic permutations. Provide proof and find the smallest \( x \) that satisfies this condition.

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