As shown by Reliable Broadcast and Consensus, cooperation among a set of independent computing entities (sequential processes) is a central issue in distributed computing. Considering $n$-process asynchronous message-passing systems where some processes can be Byzantine, this paper introduces a new cooperation abstraction denoted Context-Adaptive Cooperation (CAC). While Reliable Broadcast is a one-to-$n$ cooperation abstraction and Consensus is an $n$-to-$n$ cooperation abstraction, CAC is a $d$-to-$n$ cooperation abstraction where the parameter $d$ ($1\leq d\leq n$) depends on the run and remains unknown to the processes. Moreover, the correct processes accept the same set of $\ell$ pairs $\langle v,i\rangle$ ($v$ is the value proposed by $p_i$) from the $d$ proposer processes, where $1 \leq \ell \leq d$ and, as $d$, $\ell$ remains unknown to the processes (except in specific cases). Those $\ell$ values are accepted one at a time in different orders at each process. Furthermore, CAC provides the processes with an imperfect oracle that gives information about the values that they may accept in the future. In a very interesting way, the CAC abstraction is particularly efficient in favorable circumstances. To illustrate its practical use, the paper describes in detail two applications that benefit from the abstraction: a fast consensus implementation under low contention (named Cascading Consensus), and a novel naming problem.