
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
Define a sequence by setting $x_0=7$ and using the recurrence relation
\[
x_{n+1} = \frac{x_n^2 + 4x_n + 3}{x_n + 7}
\]
for all non-negative integers $n$. Let $m$ be the smallest positive integer such that
\[
x_m \leq 3 + \frac{1}{2^{30}}.
\]
Determine in which of the following intervals $m$ lies.
A) $[100, 200]$  
B) $[201, 300]$  
C) $[301, 400]$  
D) $[401, 500]$  
E) $[501, \infty)$

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