Squares and Square-Roots, Galore!

Algebra Level 5

Suppose we have two sequences \(a_1,a_2,\ldots a_{2014}\) and \(b_1,b_2,\ldots b_{2014}\) such that all terms in both sequences are in the range \([0,1]\). Let \(M\) be the maximum value of \[\left(\sum_{i=1}^{2014}\sqrt{a_i^2+b_i^2}\right)-\sqrt{\left(\sum_{i=1}^{2014}a_i\right)^2+\left(\sum_{i=1}^{2014}b_i\right)^2}\] What is \(\lfloor M\rfloor\)?

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