Real function

Algebra Level 5

You are given that \(a,b\) and \(c\) are positive real numbers satisfying \(a+b+c\leq \frac{3}{2}\). Let \(P\) is the minimum value of the expression below.

\[ \sum_{\text{cyclic}} \sqrt{2a^2+\frac{1}{a^2b^2}+\frac{2}{b}}\] Find \(\left \lfloor 1000P \right \rfloor\)

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