A10783 | Bitwise Formula

The first line contains two integers $n$ and $m$ , the number of variables and bit depth, respectively ( $1<=n<=5000$ ; $1<=m<=1000$ ).

The following $n$ lines contain descriptions of the variables. Each line describes exactly one variable. Description has the following format: name of a new variable, space, sign ":=", space, followed by one of:

1. Binary number of exactly $m$ bits.
2. The first operand, space, bitwise operation ("AND", "OR" or "XOR"), space, the second operand. Each operand is either the name of variable defined before or symbol '?', indicating the number chosen by Peter.

Variable names are strings consisting of lowercase Latin letters with length at most 10. All variable names are different.

In the first line output the minimum number that should be chosen by Peter, to make the sum of all variable values minimum possible, in the second line output the minimum number that should be chosen by Peter, to make the sum of all variable values maximum possible. Both numbers should be printed as $m$ -bit binary numbers.

In the first sample if Peter chooses a number $011_{2}$ , then $a=101_{2},b=011_{2},c=000_{2}$ , the sum of their values is $8$ . If he chooses the number $100_{2}$ , then $a=101_{2},b=011_{2},c=111_{2}$ , the sum of their values is $15$ .

For the second test, the minimum and maximum sum of variables $a$ , $bb$ , $cx$ , $d$ and $e$ is 2, and this sum doesn't depend on the number chosen by Peter, so the minimum Peter can choose is $0$ .