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chew the/(one's) cud
To contemplate something. ("Cud" is partially digested food that is regurgitated to be chewed again, a common behavior of cows.) I've been chewing my cud for days, but I still haven't decided whether I'm taking the job or not. Sometimes you just need to stop chewing the cud and make a decision.
chew one's cud
Fig. to think deeply; to be deeply involved in private thought. (Alludes to the cow's habit of bringing food back from the first stomach into the mouth to chew it, called chewing the cud.) He's chewing his cud about what to do next.
chew the cud(informal)
to think about something carefully and for a long time He sat chewing the cud all morning.
chew the cud
Also, chew over. Ponder over, meditate, as in John tends to chew the cud before he answers, or Let me chew that over and let you know. The first term, first recorded in 1382, transfers the appearance of a patiently ruminating cow to a person deep in thought. The variant was first recorded in 1696.
chew the cudSlang
To ponder over; meditate.