Genius's Gambit
Description
You are given three integers $a$, $b$, $k$.
Find two binary integers $x$ and $y$ ($x \ge y$) such that
- both $x$ and $y$ consist of $a$ zeroes and $b$ ones;
- $x - y$ (also written in binary form) has exactly $k$ ones.
The only line contains three integers $a$, $b$, and $k$ ($0 \leq a$; $1 \leq b$; $0 \leq k \leq a + b \leq 2 \cdot 10^5$) — the number of zeroes, ones, and the number of ones in the result.
If it's possible to find two suitable integers, print "Yes" followed by $x$ and $y$ in base-2.
Otherwise print "No".
If there are multiple possible answers, print any of them.
Input
The only line contains three integers $a$, $b$, and $k$ ($0 \leq a$; $1 \leq b$; $0 \leq k \leq a + b \leq 2 \cdot 10^5$) — the number of zeroes, ones, and the number of ones in the result.
Output
If it's possible to find two suitable integers, print "Yes" followed by $x$ and $y$ in base-2.
Otherwise print "No".
If there are multiple possible answers, print any of them.
Samples
4 2 3
Yes
101000
100001
3 2 1
Yes
10100
10010
3 2 5
No
Note
In the first example, $x = 101000_2 = 2^5 + 2^3 = 40_{10}$, $y = 100001_2 = 2^5 + 2^0 = 33_{10}$, $40_{10} - 33_{10} = 7_{10} = 2^2 + 2^1 + 2^0 = 111_{2}$. Hence $x-y$ has $3$ ones in base-2.
In the second example, $x = 10100_2 = 2^4 + 2^2 = 20_{10}$, $y = 10010_2 = 2^4 + 2^1 = 18$, $x - y = 20 - 18 = 2_{10} = 10_{2}$. This is precisely one 1.
In the third example, one may show, that it's impossible to find an answer.