What does an interior Pareto-efficient allocation looks like in this set up?

"Had a question on a mid-term paper yesterday (now submitted). One question stumped me a little bit. In particular the "interior Pareto-efficient allocation" part stumped me (I’ll explain how I answered after the question). Here’s the question:

$$ U_i(c_i)= sum_{t=1}^{∞} beta_{i}^{t-1} u_{i} (c_{it}) $$

where $$ 0 <beta_{^i} < 1 $$ and $u_{i}$ is twice continuously differentiable with $u_i^′(z) > 0$ and $u^{′′}_i (z) < 0$ and satisfies $limlimits_{z to 0} u^′_i(z) = infty$ and $limlimits_{zto infty}u^′_i(z) = 0$.

Assume that all $beta_i$‘s are distinct.

Consider that 1 unit of the good is available at each period.
Describe what interior Pareto-efficient allocation looks like, and explain why."

So, I think my answer is wrong but I wrote that every period has a Pareto efficient allocation because randomly (we don’t know how), one agent receives the good and that means their utility increases and the other n-1 agents utility doesn’t decrease. Also, there is no production and no exchange, so Pareto-interior allocation (through the form of marginal rate of utilities being equalised) doesn’t come into play?

Those are my thoughts. I was pretty stumped.

(I assumed because they were not in the question that the good is not divisible, exchange is not allowed, and that we do not know how the allocation takes place.)