
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 $a \in \mathbb{R}$, and $f(x) = \frac{a \cdot 2^x - a^{-2}}{2^x + 1}$ be an odd function.  
(1) Find the zero of the function $F(x) = f(x) + 2^x - \frac{4}{2^x + 1} - 1$.  
(2) Let $g(x) = 2\log_{2}\left(\frac{1+x}{k}\right)$. If the inequality $f^{-1}(x) \leq g(x)$ holds for all $x$ in the interval $\left[\frac{1}{2}, \frac{2}{3}\right]$, find the range of the real number $k$.

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