Triple variable sequence

Using Newton sum or See this.

We get,

$\Rightarrow xy+yz+zx=-1$

$\Rightarrow xyz=0$

Now, again using Newton sum.

General Formula: For $i \geq 3$ .

$\rightarrow a^i+b^i+c^i=(a+b+c)(a^{i-1}+b^{i-1}+c^{i-1})-(ab+bc+ca)(a^{i-2}+b^{i-2}+c^{i-2}+abc(a^{i-3}+b^{i-3}+c^{i-3})$

$\Rightarrow x^4+y^4+z^4=2×14+6=34$

$\Rightarrow x^5+y^5+z^5=2×34+14=82$

$\Rightarrow x^6+y^6+z^6=2×82+34=198$

$\Rightarrow x^7+y^7+z^7=2×198+82=\boxed{478}$