Final answer:
The text provided from Diodorus of Sicily describes an event involving Alexander the Great, not Agamemnon. However, drawing from Homeric sources, when Agamemnon arrived at his palace, he would have likely sought to be reunited with his family and reclaim his throne.
Step-by-step explanation:
When Agamemnon first arrived at the palace, his intention isn't explicitly detailed in the provided text from Diodorus of Sicily's 'Library of World History.' The passage instead describes an event involving Alexander the Great and his companions. However, if we refer to Homer's 'The Odyssey,' which includes Agamemnon's story, upon his return to Mycenae after the Trojan War, Agamemnon's primary desire was likely to be reunited with his family and to reclaim his throne, making option (a) 'Be crowned as king' the most likely activity he would pursue. Note that the passage provided does not directly answer the question about Agamemnon but discusses an incident involving Alexander the Great instead.