Let $X = A \cup B$, where $A,B$ are open simply connected subspaces such that $A \cap B$ consists of $n \geq 1$ path-connected components. Prove that $\pi_1(X) \simeq \mathcal{F}_{n-1}$, the free group with $n-1$ generators.
Intuitively i can see $X$ is "similar" to the bouquet of $n-1$ copies of $S^1$, which from Van-Kampen's theorem has the fundamental group $\mathcal{F}_{n-1}$, but i couldn't formally show any connection between the spaces (such as a deformation retraction that retracts $X$ to $\bigvee_{i=1}^{n-1} S^1$)
Any help would be appreciated.