Let $a, b, c$ be three distinct positive real numbers such that $2a^{\log_e a} = bc^{\log_e b}$ and $b^{\log_e 2} = a^{\log_e c}$. Then $6a + 5bc$ is equal to $\_\_\_\_$.
Let $a, b, c$ be three distinct positive real numbers such that $2a^{\log_e a} = bc^{\log_e b}$ and $b^{\log_e 2} = a^{\log_e c}$. Then $6a + 5bc$ is equal to $\_\_\_\_$.