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Maximum run between identical elements

This is a overhaul of this now deleted question by ar kang. If the OP of that question would like to reclaim this question or has a problem with me posting this I'd be happy to accommodate

Given a list of integers as input find the maximum possible sum of a continuous sublist that starts and ends with the same value. The sublists must be of length at least 2. For example for the list

[1, 2, -2, 4, 1, 4]

There are 2 different continuous sublists start and end with the same value

[1,2,-2,4,1] -> 6
[4,1,4]      -> 9

The bigger sum is 9 so you output 9.

You may assume every input contains at least 1 duplicate.

This is code-golf so answers will be scored in bytes with fewer bytes being better.

Test cases

[1,2,-2,4,1,4]  -> 9
[1,2,1,2]       -> 5
[-1,-2,-1,-2]   -> -4
[1,1,1,8,-1,8]  -> 15
[1,1,1,-1,6,-1] -> 4
[2,8,2,-3,2]    -> 12
[1,1,80]        -> 2
[2,8,2,3,2]     -> 17

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