
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 real number and \( c \neq 0 \) an integer such that

\[
c-0.1 < x^p \left( \frac{1-(1+x)^{10}}{1+(1+x)^{10}} \right) < c+0.1
\]

for all (positive) real numbers \( x \) with \( 0 < x < 10^{-100} \). Find the ordered pair \( (p, c) \).

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