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

Algebra Level 5

What is the maximum possible value of the expression i=1200ai2ai+1,\sum^{200}_{i=1}|a_i-2a_{i+1}|, where a1,a2,,a201a_1, a_2, \ldots, a_{201} are real numbers such that 1ai41\leq a_i\leq 4 for i=1i=1 to 201?

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