
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
The fare for a minibus ride is 75 rubles. Fifteen passengers enter one by one and do not exchange money among themselves. With a probability of 0.5, a passenger gives the driver a 100-ruble bill and receives 25 rubles in change, in coins of 5 or 10 rubles, and with a probability of 0.5, a passenger gives the driver a 50-ruble bill and 25 rubles in coins.

a) Before the trip, the driver had no coins at all. 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) How much in coins should the driver have at the beginning of the trip to be able to give change to all passengers who pay with a 100-ruble bill with a probability of at least 0.95?

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