If \( a, b, c, d, e \) and \( f \) are six real numbers such that \...

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If \( a, b, c, d, e \) and \( f \) are six real numbers such that
\( \mathrm{P} \)
\[
\begin{aligned}
a+b+c & =d+e+f \\
a^{2}+b^{2}+c^{2} & =d^{2}+e^{2}+f^{2}
\end{aligned}
\]
W and \( a^{3}+b^{3}+c^{3}=d^{3}+e^{3}+f^{3} \), prove by mathematical induction that
\[
a^{n}+b^{n}+c^{n}=d^{n}+e^{n}+f^{n}, \forall n \in N \text {. }
\]
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