
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 ride in a minibus costs 75 rubles. Fifteen passengers board one by one without exchanging money with each other. Each passenger has a 0.5 probability of giving the driver a 100-ruble bill and receiving 25 rubles in change (in coins of 5 or 10 rubles), and a 0.5 probability of giving the driver a 50-ruble bill and 25 rubles in coins.

a) (Grade 9, 5 points) Initially, the driver had no coins. What is the probability that the driver will be able to immediately give change to each passenger who pays with a 100-ruble bill?

b) (Grade 9, 7 points) How much money in coins should the driver have before the trip to have at least a 0.95 probability of being able to immediately give change to all passengers who pay with a 100-ruble bill?

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