Finding Expected Value

Mr. Chanek opened a letter from his fellow, who is currently studying at Singanesia. Here is what it says.

Define an array $$$b$$$ ($$$0 \leq b_i < k$$$) with $$$n$$$ integers. While there exists a pair $$$(i, j)$$$ such that $$$b_i \ne b_j$$$, do the following operation:

• Randomly pick a number $$$i$$$ satisfying $$$0 \leq i < n$$$. Note that each number $$$i$$$ has a probability of $$$\frac{1}{n}$$$ to be picked.
• Randomly Pick a number $$$j$$$ satisfying $$$0 \leq j < k$$$.
• Change the value of $$$b_i$$$ to $$$j$$$. It is possible for $$$b_i$$$ to be changed to the same value.

Denote $$$f(b)$$$ as the expected number of operations done to $$$b$$$ until all elements of $$$b$$$ are equal.

You are given two integers $$$n$$$ and $$$k$$$, and an array $$$a$$$ ($$$-1 \leq a_i < k$$$) of $$$n$$$ integers.

For every index $$$i$$$ with $$$a_i = -1$$$, replace $$$a_i$$$ with a random number $$$j$$$ satisfying $$$0 \leq j < k$$$. Let $$$c$$$ be the number of occurrences of $$$-1$$$ in $$$a$$$. There are $$$k^c$$$ possibilites of $$$a$$$ after the replacement, each with equal probability of being the final array.

Find the expected value of $$$f(a)$$$ modulo $$$10^9 + 7$$$.

Formally, let $$$M = 10^9 + 7$$$. It can be shown that the answer can be expressed as an irreducible fraction $$$\frac{p}{q}$$$, where $$$p$$$ and $$$q$$$ are integers and $$$q \not \equiv 0 \pmod{M}$$$. Output the integer equal to $$$p \cdot q^{-1} \bmod M$$$. In other words, output such an integer $$$x$$$ that $$$0 \le x < M$$$ and $$$x \cdot q \equiv p \pmod{M}$$$.

After reading the letter, Mr. Chanek gave the task to you. Solve it for the sake of their friendship!