Hungary started general elections on Sunday to elect a 199-seat parliament, which could make the current Prime Minister Viktor Orban succeed in a third straight term.
Some 8.3 million Hungarians are eligible to vote, with preliminary results expected on Sunday night.
Analysts say the high turnout, especially in opposition-leaning Budapest and other big cities, may help other parties to overcome a mainly first-past-the-post election system which disadvantages the divided opposition.
The Fidesz Party is projected to win with a two-third majority, 134 seats, while the nationalist Jobbik Party is expected to win 27 seats, according to the election office website.
A strong victory could embolden him to put more muscle into a Central European alliance against the European Union's migration policies.
"Theoretically everything is still possible as we don't know the data yet. but in Hungary a two-thirds victory is possible if neither side loses more than 10 districts and there is a difference of at least 20 percent between the victor and the runner-up", Gulyas said.
Orban and his wife Aniko Levai voted at a polling station in a school in a Budapest suburb.
Over 3.3 million voters had taken part by 1 p.m. (1100 GMT), for a turnout rate of 42.3 percent six hours before the end of voting. "I voted for Fidesz and I am asking everyone to take part in the election", Orban told a group of dozens of journalists. He claims that the opposition is collaborating with the United Nations, the European Union and wealthy philanthropist George Soros to turn Hungary into an "immigrant country", threatening its security and Christian identity.
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While Fidesz led all opinion polls before the vote, there is a small chance that the fragmented opposition could strip Fidesz of its parliamentary majority if voters frustrated with Orban's policies choose tactical voting in the 106 constituencies.
He said voter turnout would determine the outcome.
A strong win for Orban would also boost other right-wing nationalists in Central Europe, in Poland and in neighboring Austria, and expose cracks in the 28-nation EU.
The anti-immigrant campaign has gone down well with numerous roughly two million core voters of Fidesz. Opposition parties are keen to make sure that Orban's bloc does not sweep to a supermajority in which the autocratic leader could easily push through more constitutional changes. The EU is in Berlin, in Budapest, in Warsaw, in Prague and in Bucharest.
"High turnout means, most probably, less mandates for Fidesz than in the previous term", said Peter Kreko, director of think tank Political Capital.
Heartened by a recent poll showing more Hungarians would prefer for Orban to go than to stay, the opposition has shifted its focus from coordination to get-out-the-vote campaigns.
Fidesz may even be on track to win its coveted two-thirds "super-majority", which would grant it wide powers to press ahead with controversial measures and change the constitution.