Let $G$ be a topological group and $X$ be a topological space. Let $\alpha$, $\beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous path of actions connecting $\alpha$ to $\beta$. This means that there is a continuous function $\Gamma:[0,1]\times G\times X \to X$ such that each $\Gamma_t$ is a group action and we have $\Gamma_0=\alpha, \Gamma_1=\beta$.

**Question 1:** What is an example of the following situation:
There are two group actions $\alpha$, $\beta$ by $G=S^1$ on a manifold $M$ which are not homotopic actions but for every $g\in G$, the two homeomorphisms $\alpha_g$, $\beta_g$ of $M$ are homotopic maps?

**Question 2:** What is an example of the following situation:
There are two group actions $\alpha$, $\beta$ by $ G=S^1$ on a manifold $M$ which are not homotopic actions but $\alpha, \beta$ are homotopic maps as maps from $G\times X$ to $X$?