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John likes to learn esoteric programming languages. Recently he discovered the probabilistic programming language Java2K. Built-in functions of Java2K have only a certain probability to do whatever you intend them to do.
The Java2K programming is very hard, so John designed a much simpler language for training: Java2016. Built-in operators of Java2016 are deterministic, while their operands are random. Each value in Java2016 is a positive integer in the range 0..255, inclusive.
Java2016 supports six operators of three precedencies:
〈expression〉 ::= 〈expression〉‘
min’〈sum〉 | 〈expression〉‘
max’〈sum〉 | 〈sum〉
〈sum〉 ::= 〈sum〉‘
+’〈term〉 | 〈sum〉‘
-’〈term〉 | 〈term〉
〈term〉 ::= 〈term〉‘
*’〈factor〉 | 〈term〉‘
/’〈factor〉 | 〈factor〉
〈factor〉 ::= ‘
)’ | ‘
?’ | 〈macro〉
min’) and maximum (‘
max’) operators are defined as usual. Addition (‘
+’), subtraction (‘
-’) and multiplication (‘
*’) are defined modulo 256. The result of the division (‘
/’) is rounded towards zero. If the divider is zero, the program crashes. The argument of the operator is a result of another operator, evenly distributed random value (‘
?’), or macro substitution.
For instance, the probability that “
?/?/?” is evaluated to zero is 98.2%, while the probability of the crash is 0.8%.
The Java2016 program consists of zero or more macro definitions, followed by the resulting expression. Each macro definition has a form of
〈macrodef〉 ::= 〈macro〉‘=’〈expression〉
〈macro〉 ::= ‘
a’ . . . ‘
The macro should be defined before the first use. It may not be redefined. The macro is expanded to its definition on each use. For instance,
a = ? max ?
(a max a) / a
is expanded to “
((? max ?) max (? max ?)) / (? max ?)”.
John is going to add probabilistic constants to Java2016, so for each possible constant value he needs a program that successfully evaluates to this value with at least one-half probability. Crashes are counted toward failures.
The input contains a single integer c — the target constant (0 ≤ c ≤ 255).
Output a Java2016 program that successfully evaluates to constant c with probability no less than 1/2. The total length of the program should not exceed 256 characters (excluding spaces).
? /?/ ?
a = ? max ? (a max a) / a