KöMaL 2020 A789

Let $p(x)=a_{21}x^{21}+a_{20}x^{20}+\cdots +a_1x+1$ be a polynomial with integer coefficients and real roots such that the absolute value of all of its roots are less than $1/3$, and all the coefficients of $p(x)$ are lying in the interval $[-2019a,2019a]$ for some positive integer $a$. Prove that if this polynomial is reducible in $\mathbb{Z}[x]$, then the coefficients of one its factors are less than $a$.