
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 equation ${\left({x}^{2}+2x+2\right)}^{5}={a}\_{0}+{a}\_{1}\left(x+1\right)+{a}\_{2}{\left(x+1\right)}^{2}+\dots+{a}\_{10}{\left(x+1\right)}\_{10}$, where ${{a}\_{i}} (i=1,2,\dots ,10)$ are real constants. Find:

(1) The value of $\sum\limits\_{n=1}^{10}{{{a}\_{n}}}$;
(2) The value of $\sum\limits\_{n=1}^{10}{n{{a}\_{n}}}$.

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