I always thought these types of systems are kind of weird.
There are a lot of countries where the president gets elected separately, then after parliamentary elections, the president appoints the Prime Minister based on the recommendation from the people. That recommendation of course, is the person who leads the party that "won". I put won in brackets because winning doesn't often indicate a clear mandate, especially in countries with numerous parties splitting the vote. It does seem like the president has some discretion depending on the constitution of the country, but going against decorum and tradition isn't a great practice.
It's not that different in Canada. Here, the Prime Minister is technically appointed by the governor general after the election. I believe the only requirement is that they have a seat in parliament and they have the confidence of the House of Commons. Traditionally and through convention, this has been the leader of the party that won the most seats, but theoretically, it doesn't have to be. In such a case, you could have the party leader with the second most seats become the appointed Prime Minister if enough other parties supported it.
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