
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
Let \( p \) be a positive number, and consider the infinite sequence \( 1, p, \frac{1}{p}, p^{2}, \frac{1}{p^{2}}, \ldots, p^{n}, \frac{1}{p^{n}} \ldots \). For which values of \( p \) is it true that any positive number can be approximated to any desired accuracy by the sum of a finite number of suitable elements from the above sequence?

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