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For the first time primary hot isotope distributions are experimentallyreconstructed in intermediate heavy ion collisions and used withantisymmetrized molecular dynamics (AMD) calculations to determine density,temperature and symmetry energy coefficient in a self-consistent manner. Akinematical focusing method is employed to reconstruct the primary hot fragmentyield distributions for multifragmentation events observed in the reactionsystem $^{64}$Zn + $^{112}$Sn at 40 MeV/nucleon. The reconstructed yielddistributions are in good agreement with the primary isotope distributions ofAMD simulations. The experimentally extracted values of the symmetry energycoefficient relative to the temperature, $a_{sym}/T$, are compared with thoseof the AMD simulations with different density dependence of the symmetry energyterm. The calculated $a_{sym}/T$ values changes according to the differentinteractions. By comparison of the experimental values of $a_{sym}/T$ withthose of calculations, the density of the source at fragment formation wasdetermined to be $\rho /\rho_{0} = (0.63 \pm 0.03 )$. Using this density, thesymmetry energy coefficient and the temperature are determined in aself-consistent manner as $a_{sym} = (24.7 \pm 1.9) MeV$ and $T=(4.9 \pm 0.2)$MeV

A novel determination of density, temperature and symmetry energy for nuclear multi-fragmentation through primary fragment yield reconstruction