Using a square root decomposition approach with lazy propagation, ~A~ is divided into about ~\sqrt{N}~ blocks, each with a size of approximately ~\sqrt{N}~. For each block, maintain a frequency array ~F~, where ~F_i~ ~(1 \le i \le 500)~ represents the number of books labeled ~i~ in that block. Additionally, keep track of a variable ~lazy~, initially set to ~-1~, representing whether the entire block's books all have a label of ~lazy~.

When updating a range ~(l, r)~ to ~v_i~, there will be at most ~\sqrt{N}~ blocks fully covered by the range and at most ~2~ blocks that are partially covered. For fully covered blocks, update their value ~lazy~ to ~v_i~. For partially covered blocks, if ~lazy = -1~, update ~F~ and ~A~ based on ~lazy~ (eg. ~F_{lazy} = ~ block size) and set ~lazy~ back to ~-1~. Then update the individual books. A single update takes ~\mathcal{O}(\sqrt{N})~ time.

A similar approach is used to query the range ~(l, r)~ for occurrences of ~v_i~. For fully covered blocks, if ~lazy = v_i~, add the block size to the count. Otherwise, add ~F_{v_i}~. For partially covered blocks, if ~lazy = -1~, update ~F~ and ~A~ based on ~lazy~ and set ~lazy~ back to ~-1~. Then compare the individual books with ~v_i~ and add the occurrences accordingly. A single query takes ~\mathcal{O}(\sqrt{N})~ time.