The time constant of the circuit shown is:

I found $R_{th}=\dfrac{3R}{5}$ by following method :

We short-circuit the batteries to calculate Thevenin's equivalent resistance across $C$ hence $2R\parallel 2R=R$ which is parallel with $\dfrac{R}{2}+R=\dfrac{3R}{2}\implies R_{th}=R\parallel \dfrac{3R}{2}=\dfrac{3R}{5}$

So time constant $\tau=\dfrac{3RC}{5}$

But $\tau=\dfrac{RC}{2}$, given. Where I went wrong?