A given field mouse population satisfies the differential equation

$\frac{dp}{dt}=0\mathrm{.}2p-420$

where $p$ is the number of mice and $t$ is the time in months.

$($a$)$ Find the time at which the population becomes extinct if

$p(0)=2050$

Round your answer to two decimal places.

$t_f=,$ month$(s)$

$($c$)$ Find the initial population $p_0$ if the population is to become extinct in $1$ year.

Round your answer to the nearest integer.

$p_0=,$ mice