The following page addresses, in the given sequence, were generated by a program:
1 2 3 4 1 3 5 2 1 5 4 3 2 3

This program is run on a demand paged virtual memory system, with main memory size equal to $$4$$ pages. Indicate the page references for which page faults occurs for the following page replacement algorithms.
(a) $$LRU$$
(b) $$FIFO$$

Assume that the main memory is empty initially.

Let the page reference and the working set window be c c d b c e c e a d and 4, respectively. The initial working set at time $$t = 0$$ contains the pages {a, d, e}, where a was referenced at time $$t = 0, d$$ was referenced at time $$t = -1,$$ and $$e$$ was referenced at time $$t = -2.$$ Determine the total number of page faults and the average number of page frames used by computing the working set at each reference.