It happened at 31st Street and Hoyt Avenue in Astoria just before 6 a.m.
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Authorities said one of the vehicles may have attempted to cut off the other, causing the crash.
The bus had no children on board, just the driver and a matron.
No serious injuries were reported.
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The vehicles crashed on the local streets, just off the ramp from the RFK Triborough Bridge.
Authorities say the driver of the allegedly stolen vehicle was taken into custody.
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