
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Assume that the number of offspring for every man can be  $0,1,\ldots, n$  with with probabilities  $p_0,p_1,\ldots,p_n$  independently from each other, where  $p_0+p_1+\cdots+p_n=1$  and  $p_n\neq 0$ . (This is the so-called Galton-Watson process.) Which positive integer  $n$  and probabilities  $p_0,p_1,\ldots,p_n$  will maximize the probability that the offspring of a given man go extinct in exactly the tenth generation?

Present the answer in LaTex format: \boxed{Your answer}
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