
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
A certain unit is preparing to select 4 out of 8 applicants (5 males and 3 females) to compete for 4 deputy director positions.  
$(1)$ Let $X$ be the number of females selected among the 4 chosen individuals. Find the probability distribution and the expected value of $X$.  
$(2)$ If the 4 deputy directors selected are assigned to departments A, B, C, and D, with each department receiving 1 deputy director and each deputy director being assigned to only one department, find the probability that department A is headed by a female when department B is headed by a male.

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