\(09:19:29\) and \(12:11:10\) is each an example of what we will call an Arithmetic Time. These are times in the 24 hour clock, \(H:M:S\), when the integers \(H,M,S\), in that order, form an arithmetic sequence with a non-zero common difference.
On an analogue clock, let
\( 0^\circ \leq \alpha \lt 180^\circ \) be the angle between the hour and minute hands,
\( 0^\circ \leq \beta \lt 180^\circ \) be the angle between the minute and second hands, and
\( 0^\circ \leq \gamma \lt 180^\circ \) be the angle between the second and hour hands.
What is the minimum value of \( \alpha + \beta + \gamma \) (in degrees to 1 d.p.) for an Arithmetic Time?
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