Even Roasting

A chef wants to roast $N$ pieces of meat evenly. The pieces are arranged in a line and numbered $1,2,\ldots,N$ from left to right. Initially, every piece has a cooking level of $0$.

In one operation, choose the left or right side and an integer heating power $P$ ($1 \leq P \leq 10^9$). The cooking level of the $d$-th piece from the chosen side increases by $\max (0,P-d+1$. The piece at the chosen end has .

For example, suppose there are $4$ pieces with cooking levels $[0,0,0,0]$ from left to right. Heating from the left with power $3$ increases their cooking levels by $3,2,1,0$, respectively, resulting in . The fourth piece does not become more cooked in this operation.

If we then heat from the left again with power $5$, the increases are $5,4,3,2$, resulting in $[8,6,4,2]$. Thus, the increases from each operation are added to the current cooking levels.

Find the minimum number of operations needed to make the cooking levels of all pieces equal to the same positive integer, and output operations that achieve this.

The input is given in the following format:

$T$ $\mathrm{case}_1$ $\mathrm{case}_2$

For each test case, print the minimum number of operations $Q$ on the first line.

On the next $Q$ lines, print the operations in the order they are performed. On the $i$-th line, print two space-separated integers $S_i,P_i$ ().

• $1 \leq T \leq 100\,000$.
• $1 \leq N \leq 100\,000$ in each test case.
• The sum of $N$ over all test cases is at most .