If an incumbent president is seeking a second term in office, his or her party is likely to nominate the president in question. In the U.S., presidents can serve two, 4-year terms.
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If an incumbent president is willing and qualified to run for reelection, his party does not contest his nomination, most likely because of the message that would be interpreted if the party does not support its incumbent president. However, that was not always the case. Before the Civil War, several incumbent presidents, including Franklin Pierce in 1856, sought but did not receive their parties' nominations for president.
The person who is in office and seeking re-election is the incumbant electee. If he is likely to get re-elected, he is called a shoo-in.
The word "secede" seems strange here. People say a candidate withdrew or dropped out of the race, if he stopped seeking the nomination.
2/3 of the senate
The Cabinet which is made up of the secretaries of the departments is the answer you are seeking.