Description

See if a spreadsheet has a consistent formulation.

Any given cell of a spreadsheet can have a formula that depends on the value from some other cell. Before trying to calculate the value of a cell it is important to determine if the definition of the formula for that cell is circular.

Our spreadsheet only has a limited formulation for expressions:

definition := cell "=" expression

expression := term  | expression "+" term | expression "-" term

term := factor | term "*" factor | term "/" factor

factor := number | cell | "(" expression ")"

number := digit | number digit

digit := "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9"

cell := "R" digit digit "C" digit digit

Valid cell rows and columns are 1 through 20, so that a total of 400 cells are available in our spreadsheet.

Input

The input will contain one or more lines, each representing cells that are defined in the spreadsheet.

Note that, if an expression in some cell refers to some other cell, the contents of that other cell will always be defined.There are no white space characters in the expressions.

Output

A single line is printed for each cell defined in the input(and in the same order). This should be the cell name followed by "circular" if evaluating the cell results in a circular definition (either directly or indirectly). If the cell can be evaluated without a circular definition, then the word "ok" should follow the cell.

R01C01=1
R01C02=2
R01C03=R01C01+R01C02
R02C01=(R03C02+1)*R01C03
R03C02=R02C01

R01C01 ok
R01C02 ok
R01C03 ok
R02C01 circular
R03C02 circular

Hint

Clarification

The following spreadsheet would have two circular results

R01C01=0*R01C01 R02C01=R02C01-R02C01

Point is, don't do algebraic simplifications before determining circularity.

Q. If a cell references a cell that is circular, is the cell defined as circular even though it does not reference itself?

A. Yes.

Source

Rocky Mountain 2003