zusagen
to accept
To accept an invitation or an offer; to confirm one's participation or agreement.
Ich habe für die Party am Samstag zugesagt.
I accepted the invitation for the party on Saturday.

to accept / promise / appeal
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to accept
To accept an invitation or an offer; to confirm one's participation or agreement.
Ich habe für die Party am Samstag zugesagt.
I accepted the invitation for the party on Saturday.

'Zusagen' is a separable verb. The prefix 'zu' comes at the end in a main clause; in subordinate clauses the verb stays together, and in the infinitive with 'zu', the form 'zuzusagen' is formed.
Er sagte sofort für das Projekt zu.
He immediately agreed to take part in the project.
The verb is made up of the prefix 'zu' and the verb 'sagen' (to say). The prefix 'zu' signals agreement or positive confirmation here, literally 'Ja sagen' (to say yes).
Können Sie mir bis morgen zusagen?
Can you confirm by tomorrow?
In the meaning 'gefallen' (to please / to appeal), 'zusagen' is always used with a dative object. The subject is the thing that is pleasing, and the person it appeals to is in the dative.
Die neue Wohnung sagt uns sehr zu.
The new apartment really appeals to us.
When you accept an invitation, 'zusagen' is often used directly or with the preposition 'für' (for). Typical combinations are 'eine Einladung zusagen' (to accept an invitation), 'seine Teilnahme zusagen' (to confirm one's participation), or 'für eine Feier zusagen' (to accept an invitation to a celebration).
Wir haben für die Hochzeit unserer Freunde zugesagt.
We accepted the invitation to our friends' wedding.
When 'zusagen' means 'versprechen' (to promise), it is often followed by a subordinate clause with 'dass' (that) or an infinitive construction with 'zu' (to) to describe the promise.
Der Chef sagte zu, dass wir mehr Unterstützung bekommen.
The boss promised that we would get more support.
to lie / be located
'liegen' and 'zusagen' can both express that something suits a person or appeals to them. 'liegen' is used especially for activities, school subjects, or abilities: something 'liegt mir' if it fits my nature or comes easily to me. 'zusagen' more often means that I like an offer, a plan, a style, or an idea and that it matches what I have in mind. 'Das liegt mir' sounds more like ability or personal inclination. 'Das sagt mir zu' sounds more like liking or approval.

Der ruhige Urlaubsort sagt ihm mehr zu als eine große Stadt.
The quiet vacation spot appeals to him more than a big city.
Organisieren liegt Anna, deshalb plant sie die Feier.
Organizing suits Anna, so she is planning the party.