1366. Brackets

The operation of subtraction is not associative, e.g. (5-2)-1=2, but 5-(2-1)=4, therefore (5-2)-1<>5-(2-1). It implies that the value of the expression of the form 5-2-1 depends on the order of performing subtractions. Usually in lack of brackets we assume that the operations are performed from left to right, i.e. the expression 5-2-1 is equivalent to the expression (5-2)-1.

We are given an expression of the form

x1 +/- x2 +/- … +/- xn,

where each +/- denotes either + (plus) or - (minus), and x1,x2,…,xn denote (pairwise) distinct variables. In an expression of the form

x1-x2-…-xn

we want to insert n-1 pairs of brackets to unambiguously determine the order of performing subtractions and, in the same time, to obtain an expression equivalent to the given one. For example, if we want to obtain an expression equivalent to the expression

x1-x2-x3+x4+x5-x6+x7

we may insert brackets into

x1-x2-x3-x4-x5-x6-x7

as follows:

(((x1-x2)-((x3-x4)-x5))-(x6-x7)).

Note: We are interested only in fully and correctly bracketed expressions. An expression is fully and correctly bracketed when it is

either a single variable,

or an expression of the form (w1-w2), in which w1 and w2 are fully and correctly bracketed expressions.

Informally speaking, we are not interested in expressions containing spare brackets like: (), (xi), ((…)). But the expression x1-(x2-x3) is not fully bracketed because it lacks the outermost brackets.

Task

Write a program which:

1.reads the description of the given expression of the form x1 +/- x2 +/- … +/- xn,

2.computes the number of different ways (modulus 1 000 000 000) in which n-1 pairs of brackets may be inserted into the expression x1-x2-…-xn so as to unambiguously determine the order of performing subtractions and, in the same time, to obtain an expression equivalent to the given one,

3.writes the result.