calculus - Sum $sumlimits_{n=1}^inftyfrac{H_n^2}{n^22^n}$

Where $ H_n$ is the harmonic number, $\ \displaystyle H_n=1+\frac12+\frac13+...+\frac1n$.

I am going to present my solution as I need it as a reference.

Other approaches are appreciated.

here is the closed form $$\sum_{n=1}^\infty\frac{H_n^2}{n^22^n}=-\frac1{24}\ln^42+\frac14\ln^22\zeta(2)-\frac74\ln2\zeta(3)+\frac{37}{16}\zeta(4)-\operatorname{Li}_4\left(\frac12\right)$$