Conditional Statements

There is a very small programming language built from just these three statements.

if <boolean expression> then <statement list> fi
if <boolean expression> then <statement list> else <statement list> fi
checkpoint

Every keyword is written in lowercase. A <statement list> is one or more statements, and the whole program is a single <statement list>.

A <boolean expression> is made of the following.

• Variable: a single uppercase letter
• Operators: unary ~ (NOT), binary & (AND), binary | (OR)
• Precedence: ~ is highest, then &, then |.
• Parentheses for grouping.
• No spaces inside an expression.

The grammar of an expression is:

<BE> -> ~<BE> | <BE>&<BE> | <BE>|<BE> | (<BE>) | [A-Z]

Given a syntactically correct program, write a program that determines, for each checkpoint, the value every variable must take in order to reach it.

A single syntactically correct program is given on standard input.

• Keywords are always lowercase, and every boolean variable is a single uppercase letter.
• Keywords are separated by whitespace (space, tab, newline).
• An expression contains no whitespace.
• Each <statement list> contains at least one statement.
• The program uses at most 20 variables.
• The program contains at most 5000 statements.
• Every boolean expression is at most 128 characters long.

Print one line for each checkpoint in the order it appears in the program. On each line:

• First print >.
• If the checkpoint is reachable, print the variables whose value is forced in order to reach it. A variable that must be true is printed uppercase, a variable that must be false is printed lowercase, and they are listed in alphabetical order. Variables that may be either true or false are not printed.
• If the checkpoint is not reachable, print unreachable after the >.