CSES - Divisor Analysis
  • Time limit: 1.00 s
  • Memory limit: 512 MB
Given an integer, your task is to find the number, sum and product of its divisors. As an example, let us consider the number $12$:
  • the number of divisors is $6$ (they are $1$, $2$, $3$, $4$, $6$, $12$)
  • the sum of divisors is $1+2+3+4+6+12=28$
  • the product of divisors is $1 \cdot 2 \cdot 3 \cdot 4 \cdot 6 \cdot 12 = 1728$
Since the input number may be large, it is given as a prime factorization.

Input

The first line has an integer $n$: the number of parts in the prime factorization.

After this, there are $n$ lines that describe the factorization. Each line has two numbers $x$ and $k$ where $x$ is a prime and $k$ is its power.

Output

Print three integers modulo $10^9+7$: the number, sum and product of the divisors.

Constraints
  • $1 \le n \le 10^5$
  • $2 \le x \le 10^6$
  • each $x$ is a distinct prime
  • $1 \le k \le 10^9$
Example

Input:
2
2 2
3 1


Output:
6 28 1728