Breezy Mode Problem

For an array $B$, let $f(B) = $ the number of occurrences of the mode$^\dagger$ in $B$. For example, $f([2]) = 1, f([2, 3, 3]) = 2, f([3, 3, 3, 4, 4, 4]) = 3$. You’re given an array $A$ of length $N$ and $Q$ queries of the following type: - $P$ $X$: Given integers $P$ and $X$, - Set $A_P = X$; - Find the **minimum** value of $\displaystyle{\sum^{N}_{i=1} f([A_1,A_2,\ldots,A_i])}$ for any **