The family of languages described in your query

$$

$$

$$

$$

A

TM

n

$$

consists of pairs

$($

$$

$,$

$$

$)$

$($M$,$w$),$ where

$$

M is a Turing machine $($TM$)$ that accepts the string

$$

w within

$2$

$$

$2$

n

steps. This defines the behavior of the Turing machine on the input

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w with a specific time constraint.