2009 AIME II Problem 2

Suppose that a, b, and c are positive real numbers such that a^{\log _{3} 7}=27, b^{\log _{7} 11}=49, and c^{\log _{11} 25}=\sqrt{11}. Find $$
a^{\left(\log _{3} 7\right)^{2}}+b^{\left(\log _{7} 11\right)^{2}}+c^{\left(\log _{11} 25\right)^{2}}