  In this work we develop an algorithmic procedure for associating a function
defined on the Riemann surface of the $\log$ to given asymptotic data from a
function at an essential singularity. We do this by means of rational
approximations (Pad\'e approximants) used in tandem with Borel-\'Ecalle
summation. Our method is capable of handling situations where classical methods
either do not work or converge very slowly eg. \cite{DunLutz}. We provide a
general outline of the procedure and then apply it to generating approximate
tritronqu\'ee solutions to Painlev\'e's first equation ($\text{P}_\text{I}$).
Our approximations (including $\text{P}_\text{I}$) are written as a finite
linear combination of exponential integrals $\text{Ei}^+$. Furthermore, from
arXiv:2210.17502 we have explicit rational approximations for each
$\text{Ei}^+$ and thus for the approximation as a whole. In addition to
rational approximations of $\text{P}_\text{I}$, we provide the first hundred or
so poles of a tritronqu\'ee solution with essentially arbitrary accuracy which
is dependent upon the order of Pad\'e used.
