To overlook or observe closely
observe
I think you mean scrutiny. If so, it means to evalute or examine or watch carefully.
The root word of "carefully" is "care," which means to be concerned or attentive.
A word that means to read very carefully is peruse.
Watch that toddler very carefully.
The word is "diplomatic," which means choosing words carefully to avoid causing offense or misunderstandings.
duly
precise
It means to to to watch, or to carefully await something with some purpose.
Watch carefully how the magician will misdirect the attention of the audience.
"Observe" means to watch something happen.
Gently Gently