During reflex action information does not reach the brain . It goes to spinal cord. But after the reflex action the message is send to the brain and it is stored their. That is the reason when we touch a hot object we immediatly take our hand out and then never we touch the hot object again.
In reflex action brain has no role to play. it is a sudden reaction which does not involve any type of thinking. Spinal cord controls reflex action. Though the signals are sent to the brain by the nerves, but that is only to analyze them. Reflex actions take place without interference of brain.
The definition of reflex arc is correct that is there is no involvement of brain.