Unique functional $F$ on $\ell_q(\mathbb{N})$ extending $f$ and satisfying $\|F\|=\|f\|$

Question

Let $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$, $1< q<\infty$ and let $M=\{(a,b,c,0,\ldots):a,b,c\in \mathbb{F})$ be as subspace of $(\ell_q(\mathbb{N}),\|\cdot\|_q)$. Let $f:M\rightarrow\mathbb{F}$ by $(a,b,c,0,\ldots)\mapsto a-b-c$. Show that there exists a unique linear functional $F$ on $\ell_q(\mathbb{N})$ extending $f$ and satisfying $\|F\|=\|f\|$.

I know that we can use the Hahn-Banach theorem to show existence, however I'm having trouble with showing uniqueness.

I have already showed that $f$ is bounded on $(M,\|\cdot\|_q)$, and found that $\|f\|=3^{1-\frac{1}{q}}$.