Spongebob and Squares

题意翻译

我们称一个 $n \times m$ 的矩形为 —— 由 $n$ 行 $m$ 列个 $1 \times 1$ 的小正方形组成的一个矩形。 本题中给定一个整数 $x(1 \le x \le 10^{16})$，你需要求解存在多少个不同的矩形，这些矩形中恰好有 $x$ 个不同的正方形。 举个例子，假设有一个 $3 \times 5$ 的矩形，则其中存在着 $15$ 个边长为 $1$ 的正方形，$8$ 个边长为 $2$ 的正方形，$3$ 个边长为 $3$ 的正方形，所以 $3 \times 5$ 的矩形中恰好有 $15+8+3=26$ 个不同的正方形。

题目描述

Spongebob is already tired trying to reason his weird actions and calculations, so he simply asked you to find all pairs of n and m, such that there are exactly $ x $ distinct squares in the table consisting of $ n $ rows and $ m $ columns. For example, in a $ 3×5 $ table there are $ 15 $ squares with side one, $ 8 $ squares with side two and $ 3 $ squares with side three. The total number of distinct squares in a $ 3×5 $ table is $ 15+8+3=26 $ .

输入输出格式

输入格式

The first line of the input contains a single integer $ x $ ( $ 1<=x<=10^{18} $ ) — the number of squares inside the tables Spongebob is interested in.

输出格式

First print a single integer $ k $ — the number of tables with exactly $ x $ distinct squares inside. Then print $ k $ pairs of integers describing the tables. Print the pairs in the order of increasing $ n $ , and in case of equality — in the order of increasing $ m $ .

输入输出样例

输入样例 #1

26

输出样例 #1

6
1 26
2 9
3 5
5 3
9 2
26 1

输入样例 #2

2

输出样例 #2

2
1 2
2 1

输入样例 #3

8

输出样例 #3

4
1 8
2 3
3 2
8 1

说明

In a $ 1×2 $ table there are $ 2 $ $ 1×1 $ squares. So, $ 2 $ distinct squares in total. In a $ 2×3 $ table there are $ 6 $ $ 1×1 $ squares and $ 2 $ $ 2×2 $ squares. That is equal to $ 8 $ squares in total.