A comparison between products of second, third, fifth and sixth power sums

x,y,z is real numbers, prove that: $(x^2 + y^2 + z^2)(x^6 + y^6 + z^6) \geq (x^5 + y^5 + z^5)(x^3 + y^3 + z^3)$

BQ We also can prove this with CS as follows $(x^2 + y^2 + z^2)(x^4 + y^4 + z^4)\ge(x^3 + y^3 + z^3)^2; \ \ (1)$ $(x^4 + y^4 + z^4)(x^6 + y^6 + z^6)\ge(x^5 + y^5 + z^5)^2; \ \ (2)$ $(x^5 + y^5 + z^5)(x^3 + y^3 + z^3)\ge(x^4 + y^4 + z^4)^2; \ \ (3)$ Again one can multiply (1); (2); (3) to get the desired result.

Try to apply these types of inequalities before complete expansion, nobody would read that type of a proof . (No offence)

Source: InternLM Lean-Workbook, record lean_workbook_6535 (Apache-2.0). Complete source proposition preserved; proof developed independently.