
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
There are $n$ married couples at a ball. In each couple, the husband and wife are of exactly the same height, but no two couples share the same height. A waltz starts playing, and everyone pairs up randomly: each gentleman dances with a randomly chosen lady. Find the expected value of the random variable $X$ which represents the "Number of gentlemen who are shorter than their partners".

Present the answer in LaTex format: \boxed{Your answer}
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