
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 $f_n(x) = \sum\limits_{i=1}^{n-1} C_{n}^{n-i}x(x+1)...(x+i-1)$ and $g_n(x) = C_{n}^{n} + x(x+1)...(x+n-1)$, where $x \in \mathbb{R}$, $n \in \mathbb{N}^*$ and $n \geq 2$.
(1) Find the value of $n$ when $f_n(1) = 7g_n(1)$;
(2) For each given positive integer $n$, find the set of all solutions to the equation $f_n(x) + g_n(x) = 0$.

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