Robust Variance of a Random Sum (Wald's Variance Identity)

A shop receives $N$ customer orders per day, where $N \sim \operatorname{Poisson}(8)$. Each order has an independent random dollar amount $X_i$ with $E[X_i] = 50$ and $\operatorname{Var}(X_i) = 400$. Let $S = X_1 + X_2 + \cdots + X_N$ be the total daily revenue.

Using the law of total variance, derive a formula for $\operatorname{Var}(S)$ and evaluate it.

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.