First, calculate the normal reaction \(R\) on the particle:

\(R = 3g \cos 20^{\circ}\)

Next, calculate the frictional force \(F\) using the coefficient of friction \(\mu = 0.35\):

\(F = 0.35 \times 3g \cos 20^{\circ}\)

For the least possible value of \(P\), resolve forces parallel to the plane:

\(P_1 + F = 3g \sin 20^{\circ}\)

Solving for \(P_1\):

\(P_1 = 3g \sin 20^{\circ} - F\)

Substitute the values to find \(P_1 = 0.394\) (as given).

For the greatest possible value of \(P\), resolve forces in the opposite direction:

\(P_2 = F + 3g \sin 20^{\circ}\)

Substitute the values to find \(P_2 = 20.1 \text{ N}\).