
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
With the popularity of cars, the "driver's license" has become one of the essential documents for modern people. If someone signs up for a driver's license exam, they need to pass four subjects to successfully obtain the license, with subject two being the field test. In each registration, each student has 5 chances to take the subject two exam (if they pass any of the 5 exams, they can proceed to the next subject; if they fail all 5 times, they need to re-register). The first 2 attempts for the subject two exam are free, and if the first 2 attempts are unsuccessful, a re-examination fee of $200 is required for each subsequent attempt. Based on several years of data, a driving school has concluded that the probability of passing the subject two exam for male students is $\frac{3}{4}$ each time, and for female students is $\frac{2}{3}$ each time. Now, a married couple from this driving school has simultaneously signed up for the subject two exam. If each person's chances of passing the subject two exam are independent, their principle for taking the subject two exam is to pass the exam or exhaust all chances. 

$(Ⅰ)$ Find the probability that this couple will pass the subject two exam in this registration and neither of them will need to pay the re-examination fee. 

$(Ⅱ)$ Find the probability that this couple will pass the subject two exam in this registration and the total re-examination fees they incur will be $200.

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