sich beteiligen an
to participate in
To actively take part in a joint action, project, discussion, or event.
Alle Schüler haben sich an der Aufräumaktion beteiligt.
All the students participated in the cleanup campaign.

to participate / involve
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to participate in
To actively take part in a joint action, project, discussion, or event.
Alle Schüler haben sich an der Aufräumaktion beteiligt.
All the students participated in the cleanup campaign.

The verb contains the word 'Teil' (part). The prefix 'be-' and the ending '-igen' turn it into a verb. So 'sich beteiligen' literally means to make oneself part of something.
Er wollte ein Teil der Lösung sein und beteiligte sich an der Diskussion.
He wanted to be part of the solution and participated in the discussion.
This verb has two main patterns: 'sich beteiligen an' (reflexive) means to participate oneself. 'Jemanden beteiligen an' (transitive with an accusative object) means to give another person a share.
Die Bürger beteiligten sich aktiv an der Planung des neuen Parks.
The citizens actively participated in the planning of the new park.
In both meanings, 'beteiligen' is used with the fixed preposition 'an' and the dative. A common mistake is using a different preposition.
Viele Freiwillige beteiligten sich an der Hilfsaktion.
Many volunteers participated in the aid campaign.
to exclude / rule out
'sich beteiligen' and 'ausschließen' form a clear contrast here. Someone who 'sich beteiligt' actively takes part in a shared activity. Someone who is 'ausgeschlossen' is specifically not allowed to take part or be there. So 'sich beteiligen' describes participation and active involvement. 'ausschließen' describes keeping someone away from a group, place, or activity.

Samir beteiligt sich in der Bibliothek an der Planung für das Stadtfest.
Samir is participating in the planning for the town festival at the library.
Der Verein hat den Spieler nach dem Streit vom Training ausgeschlossen.
After the argument, the club excluded the player from training.