Let $f(x)$ be a polynomial with integer coefficients such that

$\begin{aligned} f(1) &= 1 \\ f(2) &= 8 \\ f(3) &= 27 \\ f(5) &= 125 \\ f(6) &= 216 \\ f(7) &= 343. \end{aligned}$

What is the minimum possible value of $|f(4)|?$

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