If $abc=1$, does $(a^2+b^2+c^2)^2 \geq a+b+c$? This question comes in mind while solving another question.

If $abc=1$, does $(a^2+b^2+c^2)^2 \geq a+b+c$ ?

I wonder if this question (if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$) helps?
I wondered if AM-GM could help, but the extra square keeps bothering me while solving it.
Another thought: If $(a^2+b^2+c^2)^2 \geq 1$, then this statement will be true. But how can I prove it?

By Rearrangement inequality we have that
$$(a^2+b^2+c^2)^2\ge a^4+b^4+c^4\ge  a^2bc+b^2ac+c^2ab=abc(a+b+c)=a+b+c$$
