AUSTRIAN-POLISH 1985 Problem 2

Suppose that $n\ge 8$ persons $P_1,P_2,\dots,P_n$ meet at a party. Assume that $P_k$ knows $k+3$ persons for $k=1,2,\dots,n-6$ . Further assume that each of $P_{n-5},P_{n-4},P_{n-3}$ knows $n-2$ persons, and each of $P_{n-2},P_{n-1},P_n$ knows $n-1$ persons. Find all integers $n\ge 8$ for which this is possible.

(It is understood that "to know" is a symmetric nonreflexive relation: if $P_i$ knows $P_j$ then $P_j$ knows $P_i$ ; to say that $P_i$ knows $p$ persons means: knows $p$ persons other than herself/himself.)