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Let $\mathscr{S}$ be the set of points inside a given equilateral triangle $A B C$ with side 1 or on its boundary. For any $M \in \mathscr{S}, a_M, b_M, c_M$ denote the distances from $M$ to $B C, C A, A B$, respectively. Define
$f(M)=a_M^3\left(b_M-c_M\right)+b_M^3\left(c_M-a_M\right)+c_M^3\left(a_M-b_M\right) .$
(a) Describe the set $\{M \in \mathscr{J} \mid f(M) \geq 0\}$ geometrically.
(b) Find the minimum and maximum values of $f(M)$ as well as the points in which these are attained.
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