Let $n$ be a positive integer with $k$ digits. A number $m$ is called an \( alero \) of \( n \) if there exist distinct digits \( a_1, a_2, \dotsb, a_k \), all different from each other and from zero, such that \( m \) is obtained by adding the digit \( a_i \) to the \( i \)-th digit of \( n \), and no sum exceeds 9.

For example, if \( n = 2024 \) and we choose \( a_1 = 2 \), \( a_2 = 1 \), \( a_3 = 5 \), \( a_4 = 3 \), then \( m = 4177 \) is an alero of \( n \), but if we choose the digits \( a_1 = 2 \), \( a_2 = 1 \), \( a_3 = 5 \), \( a_4 = 6 \), then we don't obtain an alero of \( n \), because \( 4 + 6 \) exceeds 9.

Find the smallest \( n \) which is a multiple of \( 2024 \) that has an alero which is also a multiple of \( 2024 \).

Let's think step by step and output the final answer within \boxed{}.
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