
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  $c\neq 0$  such that \[c-0.1<x^p\left(\dfrac{1-(1+x)^{10}}{1+(1+x)^{10}}\right)<c+0.1\] for all (positive) real numbers  $x$  with  $0<x<10^{-100}$ .  (The exact value  $10^{-100}$  is not important.  You could replace it with any "sufficiently small number".)

Find the ordered pair  $(p,c)$ .

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