
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: If functions $f(x)$ and $g(x)$ are differentiable on $R$, and $f(x) = g(x)$, then ${f'}(x) = {g'}(x)$. Also, British mathematician Taylor discovered an identity $e^{2x} = a_{0} + a_{1}x + a_{2}x^{2} + \ldots + a_{n}x^{n} + \ldots $. Find $a_{0} =$ ____ and $\sum_{n=1}^{10}{\frac{{a_{n+1}}}{{n{a_n}}}} =$ ____.

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