Classical DP Problem

Let us consider a grid of squares with $n$ rows and $n$ columns. Arbok has cut out some part of the grid so that, for each $i = 1, 2, \ldots, n$, only the leftmost $a\_i$ squares are remaining in the $i$-th row from the top. The values of $a\_i$ satisfy $a\_1 \le a\_2 \le \ldots \le a\_n$: that is, the grid looks like a Young diagram. Now, Arbok wants to place rooks into some of the remaining squares of the grid.

A rook is a chess piece that occupies one square and can move horizontally or vertically, through any number of unoccupied squares.

Let's say that a square is covered if it either contains a rook, or a rook can be moved to this square in one move.

Find $r$, the smallest number of rooks Arbok needs to place into some of the remaining squares so that every remaining square is covered. Also find $w$, the number of ways to put $r$ rooks to satisfy the same condition, modulo $998\\,244\\,353$.

The first line contains a single integer $n$, denoting the size of the grid ($1 \le n \le 5000$).

The second line contains $n$ integers $a\_1, a\_2, \ldots, a\_n$, denoting the widths of the rows left by Arbok ($1 \le a\_1 \le a\_2 \le \ldots \le a\_n \le n$).

Print two integers $r$ and $w$, denoting the smallest number of rooks Arbok needs to place so that every remaining square is covered, and the number of ways to put $r$ rooks to achieve the same, modulo $998\\,244\\,353$.

In the first example test, one rook is not enough to cover every square, but two rooks are enough, and there are six ways to place two rooks to cover every square (R denotes a rook, * denotes an empty square):

R          *          *          *          *          *  
**         R*         R*         R*         *R         ** 
*R*        R**        *R*        **R        R**        RR*