The Smallest $k$

Let $f(x)=7x^{11}+11x^{7}+9kx$ $g(x)=7x^{13}+13x^{7}+10kx$

Find the minimum positive value of $k$ for which $77| f(x)$ and $91|g(x)$ for every integer $x$

Details and Assumptions

• We use "$\mid$" to mean "divides evenly into". For example, $3\mid6$, and $12\mid132$.
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