
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 the function $f(x) = 2^x - 1$ with its inverse function denoted as $f^{-1}(x)$, and another function $g(x) = \log_4(3x + 1)$:
1. If $f^{-1}(x) \leq g(x)$, find the range $D$ of values for $x$.
2. Define the function $H(x) = g(x) - \frac{1}{2}f^{-1}(x)$. When $x \in D$, find the range of $H(x)$.

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