brüten
to incubate
A bird sitting on eggs to keep them warm so that the young can hatch.
Die Henne brütet seit drei Wochen auf ihren Eiern.
The hen has been incubating her eggs for three weeks.

to brood / incubate
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to incubate
A bird sitting on eggs to keep them warm so that the young can hatch.
Die Henne brütet seit drei Wochen auf ihren Eiern.
The hen has been incubating her eggs for three weeks.

The verb 'brüten' (to brood/incubate) is related to the noun 'die Brut' (brood), which refers to the offspring of animals. So the basic idea is the development or hatching of something new.
Der Vogel kümmert sich um seine Brut.
The bird takes care of its brood.
In the figurative meaning 'nachdenken' (to think), 'brüten' (to brood) is always used with the preposition 'über' and the accusative. You brood over a problem, a plan, or a task.
Die Studenten brüten über einer schwierigen Matheaufgabe.
The students are brooding over a difficult math problem.
When it is about heat, the subject of 'brüten' (to beat down) is usually impersonal, for example 'die Sonne' (the sun) or 'die Hitze' (the heat). It describes heavy, oppressive warmth.
Im Sommer brütet die Hitze über der Stadt.
In summer, the heat broods over the city.
In the literal meaning, 'brüten' (to incubate) is often used in connection with birds and the preposition 'auf' plus dative, for example 'auf Eiern brüten' (to sit on eggs/incubate eggs).
Ein Storch brütet auf dem Dach des alten Kirchturms.
A stork is nesting on the roof of the old church tower.
to consider
'überlegen' and 'brüten' can both mean that you are thinking about something. 'überlegen' is neutral and very common in everyday language. You use it when you are looking for a decision, checking something, or coming up with a plan. 'brüten über' is more figurative and sounds more intense: someone sits with a problem or task for a long time and does not make progress easily. The words can be confused because both describe thinking. 'überlegen' fits almost any everyday situation; 'brüten' fits more with difficult questions, tasks, or plans.

Er brütet seit einer Stunde über der Aufgabe und findet den Fehler nicht.
He has been brooding over the task for an hour and cannot find the mistake.
Frau Wagner überlegt, welchen Film sie am Abend mit einem Freund ansehen möchte.
Mrs. Wagner is considering which film she would like to watch with a friend in the evening.