$\lim_{n \to \infty} \left(\sum_{k=1}^n a^{k-1} \int_{(k-1)a}^{ka} {\dfrac{f(x)}{f(x) + f \left( \left(2k-1\right)a-x \right)}} \ dx \right) = \dfrac{7}{5}$

Given that function $f(x) > 0$, $\forall ~ x \in \mathbb{R}$ is bounded and satisfies the condition above. If $a < 1$, find $a$.