
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 $f\left( x \right)=\frac{1}{3}{{x}^{3}}-{{x}^{2}}+ax+1\left( a > 0 \right)$ has two distinct extreme points $\left({{x}\_{1}},{{x}\_{2}}\left( {{x}\_{1}} < {{x}\_{2}} \right) \right)$, determine the range of values for $\frac{{{x}\_{2}}}{1+\frac{{{x}\_{1}}}{{{x}\_{2}}}}$.

A) $\left( 0,2 \right)$
B) $\left( 1,4 \right)$
C) $\left( \frac{1}{2},2 \right)$
D) $\left( 0,4 \right)$

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