
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 $f(x) = \lg \frac{\sum_{i=1}^{n-1} i^{x} + n^{x}a}{n}$, where $a \in \mathbb{R}$, for any positive integer $n \geqslant 3$, if the inequality $f(x) > (x-1)\lg n$ has a solution 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