
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
An exam at a university consists of one question randomly selected from the $ n$  possible questions. A student knows only one question, but he can take the exam  $n$  times. Express as a function of  $n$  the probability  $p_n$  that the student will pass the exam. Does  $p_n$  increase or decrease as  $n$  increases? Compute  $lim_{n\to \infty}p_n$ . What is the largest lower bound of the probabilities  $p_n$ ?

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