Let $X$ be the subset of $\mathbb{R}^{4}$ given by $X=A \cup B \cup C \subset \mathbb{R}^{4}$, where $A, B$ and $C$ are defined as follows:

$$\begin{aligned} &A=\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{R}^{4}: x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=1\right\} \\ &B=\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{R}^{4}: x_{1}=x_{2}=0, x_{3}^{2}+x_{4}^{2} \leqslant 1\right\} \\ &C=\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{R}^{4}: x_{3}=x_{4}=0, x_{1}^{2}+x_{2}^{2} \leqslant 1\right\} \end{aligned}$$

Compute $H_{*}(X)$