AAM Actionable Time in a Target Activity

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If we are layering a target activity with AAM Audiences. Will a targeted individual see the personalized experience the first time they land on the page or the second time? When talking about "Audience Manager Audiences becoming actionable", does it relate to a new cookie or old cookies as well?

E.g for a personalized activity for a product held logic. We want a savings ad for a client who only has a chequing account. When they arrive on the landing page on the device they regularly use for their account, will they see the savings ad the first time they land on the page or will they have to return to that page the second time to see that experience?