B737 FLT CREW DEPARTS WITH A HAT IN PACK FOD TRAP AFTER MECHANIC SAYS ACFT IS GOOD TO GO.

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I WAS PERFORMING THROUGH FLT DUTIES IN ZZZ. 10 MINS PRIOR TO SCHEDULED PUSH; MECH CAME TO FLT DECK AND ASKED TO TURN OFF PACKS. HE THEN TOLD ME SOMEONE'S HAT APPEARED TO HAVE BEEN SUCKED INTO R PACK INTAKE. SHUT OFF PACKS AND CONTINUED THROUGH FLT. CAPT CAME BACK AND WE PREPARED TO PUSH. MECH STOOD ON JETWAY AND GAVE US 2 THUMBS UP. I PRESUMED INTAKE CHK WAS GOOD; TURNED PACKS ON; AND WE CONTINUED WITH PUSHBACK. JUST PRIOR TO GND DISCONNECT; CAPT NOTICED PUSHBACK DRIVER WAS MECH WHO DID THE INSPECTION AND ASKED IF THEY GOT THE HAT. MECH SAID NO; BUT THERE WERE SCREENS IN PLACE AND TRAP TO CATCH FOD AND WE WERE FINE FOR FLT. CAPT CLRED OFF GND CREW AND WE TAXIED OUT. I WAS UNSURE OF THIS SITUATION AND ASKED THE CAPT IF HE WAS COMFORTABLE WITH THE SIT. HE SAID YES AND WE CONTINUED. ENRTE I QUESTIONED WHAT HAD TAKEN PLACE AGAIN AND EXPLAINED I HAD NEVER HAD THAT SITUATION OCCUR AND STILL UNCOMFORTABLE WITH THE 'RESOLUTION.' WE DISCUSSED THE SITUATION AND DECIDED THAT MAINT IN OUR ARR CITY NEEDED TO BE NOTIFIED AND WE SHOULD HAVE QUERIED THE ZZZ MECH FURTHER. LANDED AND MAINT REMOVED PANEL TO ACCESS PACK INTAKE AND FOUND THE HAT. I DEFERRED TO THE CAPT'S EXPERIENCE WITH THE ACFT TYPE AND COMPANY OPS. I SHOULD HAVE BEEN MORE ADAMANT IN MY QUESTIONING OF THE SITUATION AND INSISTED THAT WE RETURN TO THE GATE TO HAVE MAINT DO A MORE THOROUGH INSPECTION; BECAUSE THAT WAS MY INCLINATION TO DO.