
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
Riding in a minibus costs 75 rubles. Fifteen passengers board one by one and do not exchange money among themselves. With a probability of 0.5, a passenger gives the driver a 100 ruble note 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 note and 25 rubles in coins.

a) Without any initial coins, what is the probability that the driver can immediately give change to each passenger who pays with a 100 ruble note?

b) What amount 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 paying with a 100 ruble note?

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