  We show that there is a language in $\mathsf{S}_2\mathsf{E}/_1$ (symmetric
exponential time with one bit of advice) with circuit complexity at least
$2^n/n$. In particular, the above also implies the same near-maximum circuit
lower bounds for the classes $\Sigma_2\mathsf{E}$,
$(\Sigma_2\mathsf{E}\cap\Pi_2\mathsf{E})/_1$, and
$\mathsf{ZPE}^{\mathsf{NP}}/_1$. Previously, only "half-exponential" circuit
lower bounds for these complexity classes were known, and the smallest
complexity class known to require exponential circuit complexity was
$\Delta_3\mathsf{E} = \mathsf{E}^{\Sigma_2\mathsf{P}}$ (Miltersen,
Vinodchandran, and Watanabe COCOON'99).
  Our circuit lower bounds are corollaries of an unconditional zero-error
pseudodeterministic algorithm with an $\mathsf{NP}$ oracle and one bit of
advice ($\mathsf{FZPP}^{\mathsf{NP}}/_1$) that solves the range avoidance
problem infinitely often. This algorithm also implies unconditional
infinitely-often pseudodeterministic $\mathsf{FZPP}^{\mathsf{NP}}/_1$
constructions for Ramsey graphs, rigid matrices, two-source extractors, linear
codes, and $\mathrm{K}^{\mathrm{poly}}$-random strings with nearly optimal
parameters.
  Our proofs relativize. The two main technical ingredients are (1) Korten's
$\mathsf{P}^{\mathsf{NP}}$ reduction from the range avoidance problem to
constructing hard truth tables (FOCS'21), which was in turn inspired by a
result of Je\v{r}\'abek on provability in Bounded Arithmetic (Ann. Pure Appl.
Log. 2004); and (2) the recent iterative win-win paradigm of Chen, Lu,
Oliveira, Ren, and Santhanam (FOCS'23).
