
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
João's scientific calculator has a special key that transforms any number $x$ displayed on the screen, provided that $x \neq 1$, into the number $\frac{1}{1-x}$.

a) What happens if the number 2 is displayed on the screen and we press the special key three times?

b) What happens if the number 2 is displayed on the screen and we press the special key ten times?

c) Finally, what happens if the number 2 is displayed on the screen and we press the special key 2015 times?

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