あちこちでそれを捜した。

here and there; all around; everywhere; throughout; all over; to get things in the wrong order (back to front); to become muddled up

indicates location of action; at; in; indicates time of action; indicates means of action; cause of effect; by; and then; so; indicates continuing action; indicates certainty, emphasis, etc.

that (indicating an item or person near the listener, the action of the listener, or something on their mind); it; that time; then; there (indicating a place near the listener); you

indicates direct object of action; indicates subject of causative expression; indicates an area traversed; indicates time (period) over which action takes place; indicates point of departure or separation of action; indicates object of desire, like, hate, etc.

to search (for something desired, needed); to look for; to search (for something lost); to seek (a suspect in a crime)