
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
Consider the function $f(x) = \log_a(a^x + k)$, with the domain $D$, where $a > 0$, $a \neq 1$. Given that there exist $[m, n] \subseteq D$, such that the range of $f(x)$ over $[m, n]$ is $[\frac{1}{2}m, \frac{1}{2}n]$, determine the range of $k$.

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