Bino-Ceil Harmonic series.

If the series $\sum_{n=1}^{\infty}\frac{H_{\left\lceil \frac{n}{2}\right\rceil}}{4^n n}{2n\choose n}=\sum_{n=1}^{\infty}\frac{H_n}{4^{2n-1}}\left(\frac{1/2}{an-1}-\frac{1/2}{bn-1}+\frac{1}{cn}\right)\left[{4n\choose 2n}\right]$ holds for some positive integers $a,b$ and $c$ respectively with $a$ being prime. Find the value of $a+b+c$ .

Notation: $H_n$ is nth harmonic number and $\left\lceil x \right\rceil$ is Ceiling function.

( The challenging task is to find the closed form for the aforementioned series . An original problem .)