How do I know whether the past participle of a verb needs to agree with its 'preceding direct object'?

This is an aspect of French grammar that many university students and even native speakers struggle with!Firstly some definitions. 1) The past participle (pp) of a verb is the form it takes in the perfect (past) tense. For example, in the sentence: "J'ai mangé du fromage" (I ate some cheese/ I have eaten some cheese), 'mangé' is the past participle of the verb 'manger', to eat. 2) When we talk about a past participle 'agreeing' with a verb, we mean that it behaves like an adjective as it changes depending on whether its masculine, feminine, singular or plural. 3) And last but not least, a 'preceding direct object' (dob) is an object which comes before the verb in a sentence, and which is 'receiving' the action of the verb. In the sentence "Les filles ont acheté des cadeaux" (The girls bought some presents), the presents are the direct object (but not preceding!) because they are being bought by the girls, who are the subject of the verb. We're just going to be focusing on verbs which are formed in the past tense with the verb 'avoir', as opposed to those conjugated with 'être' e.g. "Je suis allé à la piscine." If we restructured the example we used above so that it becomes "Voici les cadeaux que les filles ont achetés" (Here are the presents that the girls bought), you'll notice that an 's' has been added onto the end of the pp 'acheté'. This is because the presents (dob) come before the verb in the sentence. This rule also works if we replace the object with a pronoun, such as le, la, les, etc. For example: "J'ai vu les filles" becomes "Je les ai vues". This is not the case, however, for stressed pronouns e.g. lui, leur: "Je lui ai parlé" (I spoke to him) has no agreement.