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(cannot) possibly; (not) by any means; (not) at all; utterly; absolutely
この短期間でとうてい完成させることはできない。
There is no way to finish this within such a short period.
彼の記録はとうてい破れそうにない。
His record seems virtually impossible to break.
素人にはとうてい理解できない専門用語だ。
These are technical terms that a layperson cannot possibly understand.