Problem:
Score : 700 points

Problem Statement

Given is a sequence of N digits a_1a_2\ldots a_N, where each element is 1, 2, or 3.
Let x_{i,j} defined as follows:

x_{1,j} := a_j \quad (1 \leq j \leq N)

x_{i,j} := | x_{i-1,j} - x_{i-1,j+1} | \quad (2 \leq i \leq N and 1 \leq j \leq N+1-i)

Find x_{N,1}.

Constraints

2 \leq N \leq 10^6

a_i = 1,2,3 (1 \leq i \leq N)

Input

Input is given from Standard Input in the following format:

N
a_1a_2\ldotsa_N

Output

Print x_{N,1}.

Sample Input 1

4
1231

Sample Output 1

1

x_{1,1},x_{1,2},x_{1,3},x_{1,4} are respectively 1,2,3,1.

x_{2,1},x_{2,2},x_{2,3} are respectively |1-2| = 1,|2-3| = 1,|3-1| = 2.

x_{3,1},x_{3,2} are respectively |1-1| = 0,|1-2| = 1.

Finally, x_{4,1} = |0-1| = 1, so the answer is 1.

Sample Input 2

10
2311312312

Sample Output 2

0

Solution in Ruby (2.7.1) (judge verdict: Wrong Answer)

class ModInteger
  attr_reader :fac, :inv, :finv, :mod

  MOD = 10 ** 9 + 7

  def initialize(n, mod: MOD)
    @mod = mod
    @fac = [1, 1]
    @inv = [1, 1]
    @finv = [1, 1]

    (2..n).each do |i|
      @fac[i] = fac[i - 1] * i % mod
      @inv[i] = mod - inv[mod % i] * (mod / i) % mod
      @finv[i] = finv[i - 1] * inv[i] % mod
    end
  end

  def combination(n, k)
    return 0 if n < k
    return 0 if n < 0 || k < 0

    fac[n] * (finv[k] * finv[n - k] % mod) % mod
  end
end

class Integer
  def self.lucas(m, n, mod)
    ms = m.to_s(mod)
    ns = n.to_s(mod)
    len = [ms.size, ns.size].max

    res = 1

    1.upto(len) do |i|
      a = ms[-i].to_i
      b = ns[-i].to_i

      # p [:m, m, :n, n, :i, i, a, b, a.combination(b)]
      res *= a.combination(b)
      res %= mod
    end

    res
  end

  def combination(k)
    return 0 if self < k
    return 1 if k.zero?

    (self - k + 1..self).inject(:*) / k.factorial
  end

  def factorial
    return 1 if self.zero?

    (1..self).inject(:*)
  end
end

N = gets.to_i
A = gets.chomp.chars.map(&:to_i).each_cons(2).map { |a, b| (a - b).abs }
mi = ModInteger.new(N)

res = 0
(N - 1).times do |i|
  j = Integer.lucas(N - 2, i, 2)

  next if j == 0

  if A[i].even?
    res ^= 0
  else
    res ^= 1
  end
end

if res == 0
  puts A.include?(1) ? 0 : 2
else
  puts res
end