Easier Version of an Old HMMT Problem

Algebra Level 5

Let \(a\geq b\geq c\) be real numbers with \(a+b+c>0\) such that \[\begin{align*}a^2bc+ab^2c+abc^2+21&=a+b+c,\\a^2b+a^2c+b^2c+b^2a+c^2a+c^2b+3abc&=-3,\\a^2b^2c+ab^2c^2+a^2bc^2&=7+ab+bc+ca.\end{align*}\] If \(a^2\) can be written as \(\dfrac{m+\sqrt n}p\) for positive integers \(m,n,\) and \(p\), find \(m+n+p\).

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