
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 $f(x) = \lg \frac{1 + 2^{x} + 3^{x} + \cdots + (n-1)^{x} + a \cdot n^{x}}{n}$, where $a$ is a real number and $n$ is any given natural number with $n \geq 2$. Determine the range of values for $a$ if $f(x)$ is meaningful for $x \in (-\infty, 1]$.

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