
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
The domain of the function $f(x)$ is $C$. If it satisfies: ① $f(x)$ is a monotonic function within $C$; ② there exists $[m, n] \subseteq D$ such that the range of $f(x)$ on $[m, n]$ is $[\frac{m}{2}, \frac{n}{2}]$, then $y=f(x)$ is called a "hope function". If the function $f(x) = \log_a(a^x + t)$ ($a>0$, $a \neq 1$) is a "hope function", then the range of $t$ is \_\_\_\_\_.

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