
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
Let the function $f(x) = \ln \left( \frac{\sum_{i=1}^{n-1}i^x + n^x a}{n} \right)$, where $a \in \mathbb{R}$, for any positive integer $n$ ($n \geq 2$). If the inequality $f(x) > (x-1)\ln n$ has solutions in the interval $[1, +\infty)$, then the range of values for the real number $a$ is \_\_\_\_\_\_.

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