
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 that the monotonically increasing odd function $f(x)$ is defined on $\mathbb{R}$, and when $x > 0$, $$f(x) = \sqrt {x}+1.$$  
(1) Find the value of $f(0)$ and the analytic expression of $f(x)$;  
(2) If $f(k \cdot 4^x - 1) < f(3 \cdot 4^x - 2^{x+1})$ holds true for any $x \in \mathbb{R}$, determine the range of the real number $k$.

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