
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 the function $f(x) = ka^x - a^{-x}$ ($a > 0$ and $a \neq 1$, $k \in \mathbb{R}$), and $f(x)$ is an odd function defined on $\mathbb{R}$.

(1) Find the value of $k$.

(2) Given $f(1) = \frac{15}{4}$, let the function $g(x) = a^{2x} + a^{-2x} - 2f(x)$, $x \in [0, 1]$, find the range of $g(x)$.

(3) Under the conditions of question (2), is there a positive integer $\lambda$ such that $f(2x) \geq \lambda \cdot f(x)$ holds for any $x \in \left[-\frac{1}{2}, \frac{1}{2}\right]$? If it exists, find all such positive integers $\lambda$; if not, explain why.

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