## Members of a species of bacteria occur in two different types: $\alpha$ and $\beta$. Individual bacteria are capable of multiplying and mutating between the types according to the following rules:

Every minute, each individual will simultaneously undergo some kind of transformation.

Each individual $A$ of type $\alpha$ will, independently, do one of the following (at random with equal probability):

clone itself, resulting in a new bacterium of type $\alpha$ (alongside $A$ who remains)

split into 3 new bacteria of type $\beta$ (replacing $A$)

Each individual $B$ of type $\beta$ will, independently, do one of the following (at random with equal probability):

spawn a new bacterium of type $\alpha$ (alongside $B$ who remains)

die

If a population starts with a single bacterium of type $\alpha$, then it can be shown that there is a 0.07243802 probability that the population will eventually die out, and a 0.92756198 probability that the population will last forever. These probabilities are given rounded to 8 decimal places.

Now consider another species of bacteria, $S_{k,m}$ (where $k$ and $m$ are positive integers), which occurs in $k$ different types $\alpha_i$ for $0\le i< k$. The rules governing this species' lifecycle involve the sequence $r_n$ defined by:
$r_0 = 306$
$r_{n+1} = r_n^2 \bmod 10\,007$
Every minute, for each $i$, each bacterium $A$ of type $\alpha_i$ will independently choose an integer $j$ uniformly at random in the range $0\le j

### This problem involves advanced concepts from probability and combinatorics and process involving multiple stages. Due to its complexity, an exact solution or method to directly compute $P_{500,10}$ is very hard to formulate. However, we can develop a brief understanding of how this problem might be tackled.

The main component of this problem is dealing with stochastic processes where we have bacteria “deciding” what happens to them depending on their current state (their type $\alpha_i$) and the values of $r_{im+j}$ modulo 5. You would need to create a state-transition matrix where each element gives the probability of transitioning from one state to another.

For example, each bacteria of type $\alpha_i$ has a probability of 1/m to turn into any of the m new states (decided by values of q). You would then use this matrix to derive the final probability values for each state.

The second part of the problem is dealing with the sequence $r_n$, which is the square of the previous term modulo 10,007. You would need some kind of algorithm or method to efficiently deal with these large numbers and their modulus.

Once you have a handle on these two parts, the final part is to calculate the overall probability of extinction $P_{k,m}$. This would involve combining the probabilities of various possible outcomes (each of the $k$ types of bacteria dying out), possibly through some kind of dynamic programming or other recursive solution.

Due to its complexity, this problem falls well outside the range of high school mathematics and even many undergraduate mathematics programs. It involves knowledge of stochastic processes, combinatorics, number theory, and potentially algorithm-design, and would be more suited to advanced undergraduate or even graduate-level study.

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