I am trying to compute the product structure of $H^*\left(\mathbb RP^{2n+1};\mathbb Z_2\right)$ using spectral sequence from $S^1\to\mathbb RP^{2n+1}\to\mathbb CP^n$, assuming that we already know $H^k\left(\mathbb RP^{2n+1};\mathbb Z_2\right)=\mathbb Z_2$ for $0\le k\le2n+1$.
We know that, for $q=0,1$,
$$E_2^{p,q}=H^p\left(\mathbb CP^n;H^q(S^1;\mathbb Z_2)\right)=H^p(\mathbb CP^n;\mathbb Z_2)= \begin{cases} \mathbb Z_2 & p \text{ even}\le 2n \\ 0 & p\text{ odd,} \end{cases}$$
and $E^{p,q}_2=0$ when $q>1$.
Since we also know that $E^{p,q}_2$ converges to $H^*\left(\mathbb RP^{2n+1};\mathbb Z_2\right)$, we can deduce that all the boundary maps $d_2$ are zero map. Hence $E_2^{p,q}=E_{\infty}^{p,q}$.
In this case, let $\alpha\in H^0\left(\mathbb CP^n;H^1(S^1;\mathbb Z_2)\right)\cong H^1\left(\mathbb RP^{2n+1};\mathbb Z_2\right)=\mathbb Z_2$ be the generator. If we use the $E_2^{**}$-product structure, we should get
$$\alpha^2\in H^0\left(\mathbb CP^n;H^2(S^1;\mathbb Z_2)\right)=0.$$
However, this is a wrong answer since $\alpha^2$ should be a (nonzero) generator of $H^2\left(\mathbb RP^{2n+1};\mathbb Z_2\right)$. I think that I probably have some misunderstanding on the product structure of Leray spectral sequence. Can anyone point me out the mistake?