Denesting

Denesting is the process of simplifying an expression with embedded $n^{th}$ roots to obtain an equivalent expression with a lesser level of roots.

Ramanujan was a master in this matter and provided some amazing examples:
$\sqrt[3]{\sqrt[3]{2}-1}=(\sqrt[3]{\frac{1}{9}}-\sqrt[3]{\frac{2}{9}}+\sqrt[3]{\frac{4}{9}})\times \frac{1}{3}$
$\sqrt{\sqrt[3]{5}-\sqrt[3]{4}}=(\sqrt[3]{2}+\sqrt[3]{20}-\sqrt[3]{25})\times \frac{1}{3}$
$\sqrt[4]{\frac{3+2\sqrt[4]{5}}{3-2\sqrt[4]{5}}}=(3+\sqrt[4]{5}+\sqrt{5}+\sqrt[4]{125})\times \frac{1}{2}$

Can you denest the following expression : $\sqrt{\sqrt[3]{1404}-\sqrt[3]{875}}$ ?

The solution has the form $(\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c})\times \frac{p}{q}\text{ with }a<b<cp,q\text{ coprime}$

Answer format : $a,b,c,p,q$

[My timing: 35 sec]