## USACO 2012 November Contest, Gold

## Problem 1. Balanced Cow Breeds

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Problem 1: Balanced Cow Breeds [Brian Dean, 2012]
Farmer John usually brands his cows with a circular mark, but his branding
iron is broken, so he instead must settle for branding each cow with a mark
in the shape of a parenthesis -- (. He has two breeds of cows on his farm:
Holsteins and Guernseys. He brands each of his cows with a
parenthesis-shaped mark. Depending on which direction the cow is facing,
this might look like either a left parenthesis or a right parenthesis.
FJ's N cows are all standing in a row, each facing an arbitrary direction,
so the marks on the cows look like a string of parentheses of length N.
Looking at this lineup, FJ sees a remarkable pattern: if he scans from left
to right through just the Holsteins (in the order they appear in the
sequence), this gives a balanced string of parentheses; moreover, the same
is true for the Guernseys! To see if this is truly a rare event, please
help FJ compute the number of possible ways he could assign breeds to his N
cows so that this property holds.
There are several ways to define what it means for a string of parentheses
to be "balanced". Perhaps the simplest definition is that there must be
the same total number of ('s and )'s, and for any prefix of the string,
there must be at least as many ('s as )'s. For example, the following
strings are all balanced:
()
(())
()(()())
while these are not:
)(
())(
((())))
PROBLEM NAME: bbreeds
INPUT FORMAT:
* Line 1: A string of parentheses of length N (1 <= N <= 1000).
SAMPLE INPUT (file bbreeds.in):
(())
OUTPUT FORMAT:
* Line 1: A single integer, specifying the number of ways FJ can
assign breeds to cows so that the Holsteins form a balanced
subsequence of parentheses, and likewise for the Guernseys.
Since the answer might be a very large number, please print
the remainder of this number when divided by 2012 (i.e., print
the number mod 2012). Breed assignments involving only one
breed type are valid.
SAMPLE OUTPUT (file bbreeds.out):
6
OUTPUT DETAILS:
The following breed assignments work:
(())
HHHH
(())
GGGG
(())
HGGH
(())
GHHG
(())
HGHG
(())
GHGH

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