
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
Let $f(x) = \frac {1}{2^{x}+a} - \frac {1}{2}$ be an odd function defined on the real numbers domain $\mathbb{R}$, where $a$ is a real number.  
(i) Find the value of $a$;  
(ii) Determine the monotonicity of $f(x)$ (proof not required) and find the range of $f(x)$;  
(iii) If for any $x$ in the interval $[1, 4]$, the inequality $f\left(k - \frac {2}{x}\right) + f(2-x) > 0$ holds true for all times, determine the range of the real number $k$.

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