What is the smallest possible positive integer value of \(\beta,\) such that for some integers \(\alpha,\) \(\gamma,\) \(\delta,\) and \( \epsilon \), the following condition is true?

**Condition:** For every five-digit number \( \overline{abcde}\) that is a multiple of \(32,\) \( \alpha a + \beta b + \gamma c + \delta d + \epsilon e \) is also a multiple of \(32.\)

**Details and assumptions**

\( \overline{abc}\) means \( 100a + 10b + 1c\), as opposed to \( a \times b \times c\). As an explicit example, for \(a=2, b=3, c=4\), \(\overline{abc} = 234\) and not \( 2 \times 3 \times 4 = 24\).

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