A mass$-$spring$-$damper system has the following model $(2^{nd}$ order system$)$

$m{y}^{}+c{y}^{}+ky=f(t)$

When a unit step input $f(t)$ is applied to the system, the response $y(t)$ has a $10\%$ overshoot, a $2\%$ settling time of $4$ seconds, and a static gain of $2.$

a$)$ Find the natural frequency ${}_n$ and the damping ratio $$ of the system.

b$)$ Find the poles of the system.

c$)$ Find the mass $(m),$ damper $($c$),$ and spring constant $(k)$ of the system. Hint: after solving part $($a$),$ use the characteristic equation of a standard second order system and the static gain equation.