Matrix Operations (Making a Beautiful Matrix)

$N \times N$ matrix $U_N = (u_{ij})$ is defined as follows. $u_{ij}$ is the value in row $i$ and column $j$.

$u_{ij} = \begin{cases} 1 & \text{if } i \le j\\ 0 & \text{if } i > j \end{cases}$

In other words, $U_N$ has $1$ on the main diagonal and on every position above it, and $0$ everywhere else. For example:

$U_1 = \begin{pmatrix} 1 \end{pmatrix}, \; U_2 = \begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix}, \; U_3 = \begin{pmatrix} 1 & 1 & 1\\ 0 & 1 & 1\\ 0 & 0 & 1 \end{pmatrix}, \; U_4 = \begin{pmatrix} 1 & 1 & 1 & 1\\ 0 & 1 & 1 & 1\\ 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 1 \end{pmatrix}$

The following operations can be applied to this matrix.

• $1$ $r$ $p$ $q$: Choose integers $r$, $p$, $q$. Add $\frac{p}{q}$ to every element in row $r$. ($1 \le r \le N$, $-10^{12} \le p \le 10^{12}$, $1 \le q \le 10^6$)
• $2$ $c$ $p$ $q$: Choose integers $c$, $p$, $q$. Add $\frac{p}{q}$ to every element in column $c$. ($1 \le c \le N$, $-10^{12} \le p \le 10^{12}$, $1 \le q \le 10^6$)

A matrix is called beautiful if, for every row and every column, the sum of the absolute values of its elements is at most $\frac{N}{4}$. Formally, an $N$ by $N$ matrix $A = (a_{ij})$ is beautiful if the following holds for all $1 \le r \le N$ and $1 \le c \le N$.

$\sum_{j=1}^{N} \left|a_{rj}\right| \le \frac{N}{4}, \quad \sum_{i=1}^{N} \left|a_{ic}\right| \le \frac{N}{4}$

Starting from $U_N$, you can apply at most $2N$ operations to turn the matrix into a beautiful matrix. Find one such sequence of operations and output it.

The first line contains the integer $N$, the size of the matrix. ($1 \le N \le 400$)

On the first line, print the number of operations $Q$. It must satisfy $0 \le Q \le 2N$.

On each of the next $Q$ lines, print four integers separated by spaces that describe one operation. Each operation must be one of the following.

• $1$ $r$ $p$ $q$ ($1 \le r \le N$, $-10^{12} \le p \le 10^{12}$, $1 \le q \le 10^6$)
• $2$ $c$ $p$ $q$ ($1 \le c \le N$, $-10^{12} \le p \le 10^{12}$, $1 \le q \le 10^6$)

After all operations are applied in order to the matrix initialized as $U_N$, the result must be a beautiful matrix.