Squares and Square-Roots, Galore!

Algebra Level 5

Suppose we have two sequences a1,a2,a2014a_1,a_2,\ldots a_{2014} and b1,b2,b2014b_1,b_2,\ldots b_{2014} such that all terms in both sequences are in the range [0,1][0,1]. Let MM be the maximum value of (i=12014ai2+bi2)(i=12014ai)2+(i=12014bi)2\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 M\lfloor M\rfloor?

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