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Algebra Level 5

Let \(a_1, a_2, a_3, \ldots\) be a sequence of non-negative integers defined as follows: \[ \begin{cases} \begin{align} a_1 &= 0 \\ a_{k+1} &= \displaystyle \max_{i=1,2,\ldots,k} \left \{i + a_i + a_{k+1-i} \right \} \ \text{for }k \ge 1.\end{align} \end{cases} \] Find the value of \(a_{200}\).

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