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A virus spreads among a population of $1000$ people at a rate directly proportional to $1000-N(t)$, where $t$ represents the time in months and $N(t)$ represents the number of people infected after $t$ months. When $t=0$ months, there were $5$ people infected with the virus, and $2$ months later there were $20$ people infected.

Approximately how many people were infected after $10$ months?

A

$78$

B

$81$

C

$996$

D

$1000$

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