The role of a lieutenant governor can vary by state, but typically it includes several responsibilities. Among the options you provided, two common roles of a lieutenant governor are:
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To cast the tie-breaking vote in the state legislature - In many states, the lieutenant governor serves as the presiding officer of the state senate and has the authority to cast a vote in the event of a tie.
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To take over the governor’s duties if the governor is out of state - The lieutenant governor often steps in to fulfill the governor's responsibilities whenever the governor is unable to perform their duties (e.g., when the governor is out of state or incapacitated).
Given these roles, the most complete answer would involve both the tie-breaking vote and assuming the governor's duties. However, if you are looking for a singular answer, you might consider "to take over the governor’s duties if the governor is out of state" as a primary role of the lieutenant governor.