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The closest that Chelsea came to breaching the Middlesbrough defence was when Cole Palmer found himself with the goal at his mercy after pouncing on a rebound.
Enzo Fernandez's long-range effort was palmed to Palmer by goalkeeper Tom Glover, but the in-form England Under-21 star was unable to take advantage and blazed his effort over the bar.
Chelsea's continued struggles in front of goal will not have been helped by Nicolas Jackson jetting off to compete at the Africa Cup of Nations with Senegal, as Palmer started through the middle with Armando Broja's fitness being managed after injury.
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