car engine and constant power

1. The problem statement, all variables and given/known data
a cars engine produces constant power therefore the cars speed will increase at a constant acceleration.

is this true or false

2. Relevant equations
Power=w/t
work=fd
so P=mad/t

3. The attempt at a solution

so if p is constant it appears a must be constant. Therefore the statement is true

When I logically think about this situation I think it is false. I imagine myself driving down the highway in my stang turning 2500 rpm at 100 mph. Assume I stay in the same gear. Unless I raise the revs by giving it more gas I will not accelerate. When raising the engine rpm my engine makes more power….so how can I accelerate if my engine maintains a constant rpm(same power output)?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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