### Instruction: Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14,$ find the smallest possible value of $|w^3+z^3|.$.### Response: To solve this problem, I will use some algebraic and geometric properties of complex numbers.
First, I will write $w$ and $z$ in polar form, that is, $w=re^{i\theta}$ and $z=se^{i\phi}$ for some non-negative real numbers $r$ and $s$ and some angles $\theta$ and $\phi.$
Then, I will use the fact that $|w+z|=1$ to find a relation between $r$ and $s.$
I will also use the fact that $|w^2+z^2|=14$ to find another relation between $r$ and $s.$
Combining these two relations, I will try to express $r$ and $s$ in terms of a single parameter.
Next, I will use the formula for $w^3+z^3$ in polar form, and find the magnitude of this expression.
Finally, I will try to minimize this magnitude by choosing the parameter appropriately.