# Parallel Expectations

Time Limit: 1 Second Memory Limit: 32768 KB

We are to predict some facts about the behavior of a single processor designed
for running two programs in parallel. Programs are sequences of commands according
to the following grammar:

A

Before execution, programs are translated into machine language. A statement
of the form X := Y + Z is translated to the following set of machine instructions:

Mov R1, Y

Mov R2, Z

Add R1, R2

Mov X, R1

A MOV instruction copies the content of its second operand into its first operand.
An Add (Sub) instruction, adds (subtracts) its second operand from its first
operand and the result is stored in the first operand. Note that Y and Z denote
either a variable or an integer constant. Instructions generated for the command
X := Y - Z is similar to the above instructions, except that Sub command is
used instead of Add.

The processor is given two machine language programs and starts executing them
from the first instruction. In each step, it randomly selects one of the two
programs and runs the next instruction from the selected program. This continues
until one program reaches its end. In this situation, the remaining instructions
from the other one are executed sequentially to the end and the processor stops.
It is assumed that all variables are shared between two programs, but each program
has a separate register set. The goal of this program is to compute the expected
final value of all variables among all possible executions of the programs.
More precisely, we want to consider every possible execution of the two programs
and for each variable, calculate the average of its final value in different
executions. It is assumed that the initial value of all variables is zero.

## Input

The first line of the input contains a single integer t (1 <= t <= 10), the number of test cases, followed by the input data for each test case. The test cases are separated by exactly one blank line. Each test case consists of a pair of programs. Each program is written as a sequence of consecutive lines, each line containing exactly one command. Programs end with a line containing only the word END. You may assume that no variable in any program is named 'END'. There is no blank line between programs of one test case. There are at least one and at most 25 lines in each program. Total number of variables in two programs is no more than 10.

## Output

For each test case, the output should contain the expected final value of all variables in alphabetical order of variable names (digits precede letters in this order). Output for different test cases should be separated by exactly one blank line. Round the numbers in the output to 4 digits after decimal point. Do not omit trailing zeros after decimal point (e.g. write 1.2000 instead of 1.2).

## Sample Input

1 S := 1 + 3 END S := S+S END

## Sample Output

3.0000Submit

Source: Asia 2001, Tehran (Iran)