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Mentor of the Meek's ability checks the power of the other creature only as it enters. If that creature's power is 2 or less, the ability will trigger. Once the ability triggers, raising that creature's power above 2 won't affect that ability. Similarly, reducing the creature's power to 2 or less after it enters won't cause the ability to trigger.
If a creature enters with +1/+1 counters or a continuous effect such as that of Wedding Festivity will apply to the creature on the battlefield, those effects apply when checking to see if Mentor of the Meek's ability will trigger.
While resolving the triggered ability of Mentor of the Meek, you can't pay multiple times to draw more than one card.
