  In a recent breakthrough, Chen, Hirahara and Ren prove that $\mathsf{S_2E}/_1
\not\subset \mathsf{SIZE}[2^n/n]$ by giving a single-valued $\mathsf{FS_2P}$
algorithm for the Range Avoidance Problem ($\mathsf{Avoid}$) that works for
infinitely many input size $n$.
  Building on their work, we present a simple single-valued $\mathsf{FS_2P}$
algorithm for $\mathsf{Avoid}$ that works for all input size $n$. As a result,
we obtain the circuit lower bound $\mathsf{S_2E} \not\subset
{i.o.}$-$\mathsf{SIZE}[2^n/n]$ and many other corollaries:
  1. Almost-everywhere near-maximum circuit lower bound for $\mathsf{\Sigma_2E}
\cap \mathsf{\Pi_2E}$ and $\mathsf{ZPE}^{\mathsf{NP}}$.
  2. Pseudodeterministic $\mathsf{FZPP}^{\mathsf{NP}}$ constructions for:
Ramsey graphs, rigid matrices, pseudorandom generators, two-source extractors,
linear codes, hard truth tables, and $K^{poly}$-random strings.
