Showing there exists transitive point

Question

Suppose $X$ is a non-empty, second-countable Baire topological space.
Let $G$ be a countable collection of continuous functions $g_i:X \rightarrow X$, where $i=1,2,\dots$, such that the system $(X,g_i)$ is topologically transitive for all $i=1,2,\dots$.
I would like to show that there exists a point $x \in X$ such that $x$ is $g_i$-transitive for all $i=1,2,\dots$.
Since $(X,g_i)$ is topologically transitive for all $i=1,2,\dots$, we have that for any non-empty open sets $U,V$ in $X$, there exist infinitely many $n \in \mathbb{N}$ such that $g_i^n(U) \cap V \neq \emptyset$, or equivalently $U \cap g_i^{-n}(V) \neq \emptyset$. That is, for any $m \in \mathbb{N}$, there exists $n \in \mathbb{N}, n \geq m$ such that $g_i^n(U) \cap V \neq \emptyset$.
I need to show there is a point $x \in X$ such that the omega-limit set of $X$, $\omega(x)=X$. That means that for all $U \subseteq X$ non-empty and open and for all $m \in \mathbb{N}$, there exists $n \geq m$ such that $g_i^n(x)\in U$.
I am not sure how to choose this $x$.