
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
A function $f(x)$ defined on $(-1,1)$ satisfies the following conditions: (1) For any $x,y\in(-1,1)$, $f(x)+f(y)=f\left( \frac{x+y}{1+xy}\right) $; (2) When $x < 0, f(x) > 0 $.

(I) Determine the parity of the function $f(x)$ and explain the reason;

(II) Determine the monotonicity of the function $f(x)$ on $(0,1)$ and explain the reason;

(III) If $f\left( \frac{1}{5}\right)= \frac{1}{2} $, find the value of $f\left( \frac{1}{2}\right)-f\left( \frac{1}{11}\right)-f\left( \frac{1}{19}\right) $.

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