Consider $ABCD$ a square of side $1$. Points $P,Q,R,S$ are chosen on sides $AB$, $BC$, $CD$ and $DA$ respectively such that $|AP| = |BQ| =|CR| =|DS| = a$, with $a < 1$. The segments $AQ$, $BR$, $CS$ and $DP$ are drawn. Calculate the area of the quadrilateral that is formed in the center of the figure. [asy][asy] unitsize(1 cm); pair A, B, C, D, P, Q, R, S; A = (0,3); B = (0,0); C = (3,0); D = (3,3); P = (0,2); Q = (1,0); R = (3,1); S = (2,3); draw(A--B--C--D--cycle); draw(A--Q); draw(B--R); draw(C--S); draw(D--P); label("$A$", A, NW); label("$B$", B, SW); label("$C$", C, SE); label("$D$", D, NE); label("$P$", P, W); label("$Q$", Q, dir(270)); label("$R$", R, E); label("$S$", S, N); label("$a$", (A + P)/2, W); label("$a$", (B + Q)/2, dir(270)); label("$a$", (C + R)/2, E); label("$a$", (D + S)/2, N); [/asy][/asy]