
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
For each positive integer  $n$ , let  $r_n$  be the smallest positive root of the equation  $x^n = 7x - 4$ . There are positive real numbers  $a$ ,  $b$ , and  $c$  such that \[\lim_{n \to \infty} a^n (r_n - b) = c.\] If  $100a + 10b + c = \frac{p}{7}$  for some integer  $p$ , find  $p$ .

*Proposed by Mehtaab Sawhney*

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