
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]


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Read the following material: Expressing a fraction as the sum of two fractions is called expressing the fraction as "partial fractions."<br/>Example: Express the fraction $\frac{{1-3x}}{{{x^2}-1}}$ as partial fractions. Solution: Let $\frac{{1-3x}}{{{x^2}-1}}=\frac{M}{{x+1}}+\frac{N}{{x-1}}$, cross multiply on the right side of the equation, we get $\frac{{M(x-1)+N(x+1)}}{{(x+1)(x-1)}}=\frac{{(M+N)x+(N-M)}}{{{x^2}-1}}$. According to the question, we have $\left\{\begin{array}{l}M+N=3\\ N-M=1\end{array}\right.$, solving this system gives $\left\{\begin{array}{l}M=-2\\ N=-1\end{array}\right.$, so $\frac{{1-3x}}{{{x^2}-1}}=\frac{{-2}}{{x+1}}+\frac{{-1}}{{x-1}}$. Please use the method learned above to solve the following problems:<br/>$(1)$ Express the fraction $\frac{{2n+1}}{{{n^2}+n}}$ as partial fractions;<br/>$(2)$ Following the pattern in (1), find the value of $\frac{3}{{1×2}}-\frac{5}{{2×3}}+\frac{7}{{3×4}}-\frac{9}{{4×5}}+⋯+\frac{{39}}{{19×20}}-\frac{{41}}{{20×21}}$.

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