
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 $f(x)=\ln (1+ax)-\frac{2x}{x+2} \quad (a > 0)$,

(1) Find the extreme values of $f(x)$ when $a=\frac{1}{2}$.

(2) If $a\in (\frac{1}{2},1)$, the function $f(x)$ has two extreme points $({{x}_{1}},{{x}_{2}})$. Compare the size of $f({{x}_{1}})+f({{x}_{2}})$ with $f(0)$.

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