  We study the full asymptotic expansion of the monodromy data ({\it i.e.},
Stokes multipliers) for the first Painlev\'{e} transcendent (PI) with large
initial data or large pole parameters. Our primary approach involves refining
the complex WKB method, also known as the method of uniform asymptotics, to
approximate the second-order ODEs derived from PI's Lax pair with higher-order
accuracy. As an application, we provide a rigorous proof of the full asymptotic
expansion of the nonlinear eigenvalues proposed numerically by Bender,
Komijani, and Wang. Additionally, we present the full asymptotic expansion for
the pole parameters $(p_{n}, H_{n})$ corresponding to the $n$-th pole of the
real tritronqu\'{e}e solution of the PI equation as $n \to +\infty$.
