
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
Starting today, the great gambler Carlo Pietro decided to visit casinos daily. On the first day, he will bet on a game where the probability of winning is equal to $\frac{1}{2}$. On the second, third, and fourth days, he will bet on different games where the probabilities of winning are, respectively, $\frac{1}{3}, \frac{1}{4}, \frac{1}{5}$, and so on in the following days.

a) What is the probability that he has not won by the third day?

b) What is the probability that he has not won by the fifth day?

c) What is the probability that he has not won by the $2013^{\text{th}}$ day?

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