differential equations – Solving an ODE with paramters and taking the limit of the solution

I am very new to Mathematica and already spent a lot of time trying to do this but failed.

I am trying to solve an ODE:

solution = DSolve({-((m (1 + m) + 4/(9 (-2/3 + t) t)) y(t)) + 
     2 (-1/3 + t) y'(t) + (-2/3 + t) t y''(
       t) == (-4 (1 + C/2))/(9 (-2/3 + t) t), y(1) == 1, y'(1) == C}, 
  y, t)

where $m$ is a nonnegative integer and $C$ is a real number. (I am not sure how to incorporate that $m$ is a nonnegative integer in the code. )

I want to show that there exists a $C$ such that the solution is $0$ at infinity.
When I try that code:

Limit(y(t) /. solution((1)), t -> Infinity, m (Element) Integers)

it just spits out the same thing.

What should I do?
(Note that I don’t need to find that value of $C$; I just need to show that for every $m$, there is a number $C$ in which the solution vanishes at infinity. )