Benelux Algorithm Programming Contest 2014 Final I： Interesting Integers

I. Interesting Integers:

Undoubtedly you know of the Fibonacci numbers. Starting with F_1 = 1F1​=1 and F_2 = 1F2​=1,every next number is the sum of the two previous ones. This results in the sequence $1,1,2,3,5,8,13,\cdot \cdot \cdot$ .

Now let us consider more generally sequences that obey the same recursion relation

Gi =G_{i-1} +G_{i-2}Gi=Gi−1​+Gi−2​ for $i>2$

but start with two numbers G_1 \le G_2G1​≤G2​ of our own choice. We shall call these Gabonacci sequences. For example, if one uses G_1 = 1G1​=1and G_2 = 3G2​=3, one gets what are known as the Lucas numbers: $1,3,4,7,11,18,29,\cdot \cdot \cdot$ . These numbers are – apart from $1$ and $3$ – different from the Fibonacci numbers.

By choosing the first two numbers appropriately, you can get any number you like to appear in the Gabonacci sequence. For example, the number n appears in the sequence that starts with $1$ and $n-1$, but that is a bit lame. It would be more fun to start with numbers that are as small as possible, would you not agree?

Input Format

On the first line one positive number: the number of test cases, at most $100$. After that per test case:

• one line with a single integer $n$ (2 \le n \le 10^9)(2≤n≤109): the number to appear in the sequence.

Output Format

Per test case:

• one line with two integers $a$ and $b$ $(0<a\le b)$,such that,for G_1 = aG1​=a and G_2 = b, G_k = nG2​=b,Gk​=n for some $k$. These numbers should be the smallest possible, i.e., there should be no numbers a{'}a′ and b{'}b′with the same property, for which b{'} < bb′<b, or for which b{'} = bb′=b and a{'} < aa′<a.

样例输入

5
89
123
1000
1573655
842831057

样例输出

1 1
1 3
2 10
985 1971
2 7