Problem:
Score : 300 points

Problem Statement

There is a staircase with N steps. Takahashi is now standing at the foot of the stairs, that is, on the 0-th step.
He can climb up one or two steps at a time.

However, the treads of the a_1-th, a_2-th, a_3-th, \ldots, a_M-th steps are broken, so it is dangerous to set foot on those steps.

How many are there to climb up to the top step, that is, the N-th step, without setting foot on the broken steps?
Find the count modulo 1\ 000\ 000\ 007.

Constraints

1 \leq N \leq 10^5

0 \leq M \leq N-1

1 \leq a_1 < a_2 < ... < a_M \leq N-1

Input

Input is given from Standard Input in the following format:

N M
a_1
a_2
.
.
.
a_M

Output

Print the number of ways to climb up the stairs under the condition, modulo 1\ 000\ 000\ 007.

Sample Input 1

6 1
3

Sample Output 1

4

There are four ways to climb up the stairs, as follows:

0 \to 1 \to 2 \to 4 \to 5 \to 6

0 \to 1 \to 2 \to 4 \to 6

0 \to 2 \to 4 \to 5 \to 6

0 \to 2 \to 4 \to 6

Sample Input 2

10 2
4
5

Sample Output 2

0

There may be no way to climb up the stairs without setting foot on the broken steps.

Sample Input 3

100 5
1
23
45
67
89

Sample Output 3

608200469

Be sure to print the count modulo 1\ 000\ 000\ 007.

Solution in Ruby (2.3.3) (judge verdict: Time Limit Exceeded)

n, m = gets.chomp.split(" ").map(&:to_i)
a = [0]
1.upto(m) do |i|
  a[i] = gets.chomp.to_i
end

def combi2(n, k)
  $memo ||= {}
  key = "#{n},#{k}"
  return $memo[key] if $memo[key]
  num = 1
  1.upto(k) do |i|
    num = num * (n - i + 1) / i
  end
  $memo[key] = num
  num
end

def n_pattern(d)
  c = 0
  0.upto(d / 2) do |i|
    n2 = i
    n1 = d - 2 * i
    nn = n1 + n2
    c += combi2(nn, [n1, n2].min)
  end
  c
end

c = 1
cur = 0
1.upto(m) do |i|
  if a[i] == cur
    puts 0
    exit
  end
  d = a[i] - cur - 1
  c *= n_pattern(d)
  cur += d + 2
end
d = n - cur
c *= n_pattern(d)
puts c % 1_000_000_007