Attack of the Clones

Problem code: CLONES

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A boolean function is a function of the form f: Bn -> B, where B = {0, 1} and n is a non-negative integer called the arity of the function. Some Boolean functions are projections: pnk(x1, ..., xn) = xk. And given an m-ary function f, and n-ary functions g1, ..., gm, we can construct another n-ary function: h(x1, ..., xn) = f(g1(x1, ..., xn), ..., gm(x1, ..., xn)), called their composition. A set of functions closed under composition and containing all projections is called a clone. One trivial clone is a set of all boolean functions. Some of the special clones are:

  • Z is a set of 0-preserving functions: f(0, ..., 0) = 0;
  • P is a set of 1-preserving functions: f(1, ..., 1) = 1;
  • D is a set of self-dual functions: !f(x1, ..., xn) = f(!x1, ..., !xn);
  • A is a set of affine functions: the functions satisfying that if f(a1, ..., c, ..., an) = f(a1, ..., d, ..., an) then f(b1, ..., c, ..., bn) = f(b1, ..., d, ..., bn), where c and d are at some position i. This should hold for every valid i, a1, ..., an, b1, ... bn, c and d.

Now we are interested how many n-ary functions are there in some combinations of mentioned above sets. For example, for n = 2, there are exactly 8 functions in Z, 4 functions in the intersection of Z and P, 8 function in the complement of A and so on.

Input

The first line of the input file contains n - the arity of the boolean functions we are looking at. The second line contains the q - number of queries. Each of the next q lines will describe a query. The query is a set expression. The expression will contain the following characters: 'Z', 'P', 'D', 'A' denoting the sets, described above; 'v' - which is set union; '^' - which is set intersection; '!' which is complement; '\' which is set difference; and also '(' and ')' to define operations priority. Operations in brackets have higher priority. Otherwise the '!' operation has the higher priority and 'v', '^' and '\' are of the same priority. It is guaranteed that the expression will be correct. See samples for some examples of set expressions.

Constraints

1 <= n <= 100
1 <= q <= 100
The length of each expression won't exceed 100 characters.

Output

For each query in the input print how many n-ary function are in the set described by the according set expression modulo 1000003.

Example

Input:
2
6
Z
Z^P
!A
!(AvP)^D
AvZvP\A
!A^(Z\(Dv!P))

Output:
8
4
8
0
6
2
Author: spooky
Date Added: 17-04-2011
Time Limit: 1 sec
Source Limit: 50000 Bytes
Languages: ADA, ASM, BASH, BF, C, C99 strict, CAML, CLOJ, CLPS, CPP 4.0.0-8, CPP 4.3.2, CS2, D, ERL, F#, FORT, GO, HASK, ICK, ICON, JAR, JAVA, JS, LISP clisp, LISP sbcl, LUA, NEM, NICE, PAS fpc, PAS gpc, PERL, PERL6, PHP, PIKE, PRLG, PYTH, PYTH 3.1.2, RUBY, SCALA, SCM guile, SCM qobi, ST, TCL, TEXT, WSPC


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