
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 mapping $f: A \rightarrow B$, where $A = [0, 1]$ and $B = \mathbb{R}$. The rule of correspondence is $$f(x) = \log_{\frac{1}{2}}(2 - x) - \left(\frac{1}{3}\right)^x$$. For a real number $k \in B$ where there is no pre-image in set $A$, the range of values for $k$ is ______.

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