  We present a strategy to extract the position of the two $\Lambda(1405)$
poles from experimental photoproduction data measured recently at different
energies in the $\gamma p \to K^+ \pi^0 \Sigma^0 $ reaction at Jefferson Lab.
By means of a chiral dynamics motivated potential but with free parameters, we
solve the Bethe Salpeter equation in the coupled channels $\bar K N$ and $\pi
\Sigma$ in isospin I=0 and parameterize the amplitude for the photonuclear
reaction in terms of a linear combination of the $\pi \Sigma \to \pi \Sigma$
and $\bar K N \to \pi \Sigma$ scattering amplitudes in I=0, with a different
linear combination for each energy. Good fits to the data are obtained with
some sets of parameters, by means of which one can also predict the cross
section for the $K^- p \to \pi^0 \Sigma^0 $ reaction. These later results help
us decide among the possible solutions. The result is that the different
solutions lead to two poles similar to those found in the chiral unitary
approach. With the best result we find the two $\Lambda(1405)$ poles at
$1385-68i\mev$and $1419-22i\mev$.
