Lucky Stars Management

The company Lucky Stars has $N$ employees, numbered from 1 to $N$ in order of their position in the company. Bill, the President, has employee ID 1. Every employee except Bill has exactly one direct supervisor. The direct supervisor of employee $i$ is $P_i < i$. The annual income $A_j$ of each employee $j$ is an integer between $0$ and $10^9+6$.

One day Bill consulted a fortune-teller who is always right about the future. The fortune-teller said: "Someday, $K$ mistakes will be made, and Lucky Stars management will become difficult."

Each mistake is caused by exactly one employee. For $i=1,2,\ldots,N$, the probability that a mistake is caused by employee $i$ is $1/N$. Mistakes are independent, so one employee can cause several mistakes.

Bill decided to enforce personal responsibility in case the fortune-teller is right. When $K$ types of mistakes happen in one day, Bill wants to choose one employee to pay a fine. The employee is chosen by these rules.

• When an employee makes a mistake, that employee and every superior above them (the direct supervisor, the supervisor's supervisor, and so on, up to Bill) are responsible.
• Among the employees responsible for all $K$ types of mistakes, the employee with the largest number pays the fine.
• If $A_j$ is the annual salary of the employee who made the $j$-th mistake for $j=1,2,\ldots,K$, the fine is $A_1 \times A_2 \times \ldots \times A_K$.

Bill calculated the expected fine $E_i$ for employee $i$, assuming the fortune-teller is right, and reported it modulo $10^9+7$. For $i=1,2,\ldots,N$, write the expected fine as $Y_i/X_i$, where $X_i$ and $Y_i$ are coprime integers and $X_i$ is not divisible by $10^9+7$. The answer is $Y_i$ div $X_i$ modulo $10^9+7$.

Now you are given the company structure ($P_i$ for each employee) and the expected fines $E_i$ that Bill calculated. Decide whether all the given information could be correct. If so, find the minimum possible annual salary of Bill.

The first line contains two integers $N$ ($1 \le N \le 2 \cdot 10^5$) and $K$ ($1 \le K \le 10^9$, $K$ is odd). The second line contains $N-1$ integers $P_2, \ldots, P_N$ ($1 \le P_i \le i-1$), the direct supervisor of each employee $i$. If $N=1$, the second line is empty. The third line contains $N$ integers $E_i$ ($0 \le E_i \le 10^9+6$), the calculated expected fines of the employees.

If the calculated values are definitely wrong, print $-1$. Otherwise, print one integer: the minimum possible annual salary of Bill.