= Solution
The sharp <Hamilton cycle> threshold for the <Erdős-Rényi model> says that
$$
\mathbb P(G(n,p)\text{ is Hamiltonian})\longrightarrow1
$$
only above the window $np=\log n+\log\log n+o(1)$. The hypothesis therefore places $p(n)$ above that window. The <Hamiltonicity-to-pancyclicity sprinkling principle> then says that three independent $G(n,p(n))$ rounds contain every <cycle graph> $C_\ell$, $3\leq\ell\leq n$, <with high probability>: one round supplies a Hamilton cycle, while the other two supply the chords and short-cycle edges used to obtain all intermediate lengths.
The union of the three rounds has individual edge probability
$$
q=1-(1-p(n))^3\leq3p(n).
$$
By the standard monotone coupling, it is a subgraph of $G(n,3p(n))$. Since being <pancyclic graph>[pancyclic] is an increasing graph property, the required probability tends to one. If $3p(n)>1$, interpret the latter parameter as $\min\{3p(n),1\}$, in which case the conclusion is immediate.
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