  We extend the technique of asymptotic series matching to exponential
asymptotics expansions (transseries) and show that the extension provides a
method of finding singularities of solutions of nonlinear differential
equations, using asymptotic information. This transasymptotic matching method
is applied to Painlev\'e's first equation. The solutions of P1 that are bounded
in some direction towards infinity can be expressed as series of functions,
obtained by generalized Borel summation of formal transseries solutions; the
series converge in a neighborhood of infinity. We prove (under certain
restrictions) that the boundary of the region of convergence contains actual
poles of the associated solution. As a consequence, the position of these
exterior poles is derived from asymptotic data. In particular, we prove that
the location of the outermost pole $x_p(C)$ on $\RR^+$ of a solution is
monotonic in a parameter $C$ describing its asymptotics on antistokes
lines$^1$, and obtain rigorous bounds for $x_p(C)$. We also derive the behavior
of $x_p(C)$ for large $C\in\CC$. The appendix gives a detailed classical proof
that the only singularities of solutions of P1 are poles.
