
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
Given the function \[ f(x) = \frac{x^3}{3} + \frac{mx^2 + (m+n)x + 1}{2} \] has two extreme points at \[ x_1 \] and \[ x_2 \], with \[ x_1 \in (0,1) \] and \[ x_2 \in (1, +\infty) \]; the point \[ P(m, n) \] represents a plane region \[ D \]. If there exists a point within region \[ D \] on the graph of the function \[ y = \log_a{(x+4)} \] (where \[ a > 1 \]), determine the range of the real number \[ a \].

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