21. (Total Score: 16 points) Subquestion 1 is worth 4 points, Subquestion 2 is worth 6 points, Subquestion 3 is worth 6 points\\
As shown in the figure, $P - ABC$ is a regular triangular pyramid with base edge length 1. Points $D$, $E$, $F$ are on edges $PA$, $PB$, $PC$ respectively. The cross-section $DEF$ is parallel to the base $ABC$, and the sum of edge lengths of the frustum $DEF - ABC$ equals the sum of edge lengths of the pyramid $P - ABC$. (The sum of edge lengths is the sum of the lengths of all edges of a polyhedron)\\
(1) Prove: $P - ABC$ is a regular tetrahedron;\\
(2) If $PD = \frac { 1 } { 2 } PA$, find the dihedral angle $D - BC - A$. (Express the result using inverse trigonometric functions)\\
(3) Let the volume of the frustum $DEF - ABC$ be $V$. Does there exist a right parallelepiped with all edges equal and volume $V$ such that it has the same sum of edge lengths as the frustum $DEF - ABC$? If it exists, construct such a parallelepiped explicitly and provide a proof; if it does not exist, explain why.\\
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