time series with uniform residuals

Hi all, I have encountered a problem below, would love to get a solution from the community as I have no clue on how to approach this problem

Assume a time series with the following distribution.
St= St-1+(-1)t ∗ U
where U is a random variable with Uniform (0,a) distribution.
a) Derive a generalized solution to calculate probability of St > K
b) Given S0=1000 & a = 1, calculate the Probability of S50 > 1002
c) Confirm your answer for (b) using Monte-Carlo simulations. Please attach your script.
hmmmmm, looks like it works out that St is just sum product of t * U_t? t*U is uniform in [0,a*t], seems like? Some googling indicates that that would be hard to evaluate exactly, but im feeling like some asymptotics might simplify things. What are you studying that caused you to encounter such a dastardly problem? Maybe that will give a hint.