Take another look at the sentence:

Robin can't __________ anything on the counter. Money, glasses of orange juice, stacks of paper—all of these get swatted to the floor by Buster, her mischievous cat.

Robin can't put anything on the counter. Thus, you need a form of lay. After an auxiliary verb like can, you need the simple form of the verb [the infinitive without the to]. Consult the table below or review the rules.

Infinitive Definition Simple Present Simple Past Past Participle Present Participle
to lay to put something down lay(s) laid laid laying
to lie to rest or recline lie(s) lay lain lying

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