Symmetric Expressions Often Have Obvious Solutions. This One Doesn't.

Algebra Level 5

What is the maximum possible value of the expression \[\sum^{200}_{i=1}|a_i-2a_{i+1}|,\] where \(a_1, a_2, \ldots, a_{201}\) are real numbers such that \(1\leq a_i\leq 4\) for \(i=1\) to 201?

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