
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 $f(x) = \lg \frac{1+2^x+3^x+\ldots+n^x\cdot a}{n}$, where $a\in \mathbb{R}$, $n\in \mathbb{N}^*$, and $n\geq 2$.  
(1) When $n=2$, if the inequality $f(x) > \lg(x2^{x-1})$ has solutions, find the range of the real number $a$.  
(2) If $f(x)$ is defined for $x\in(-\infty, 1]$, find the range of the real number $a$.

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