
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 $(1+x)^{2n-1} = (1+x)^{n-1}(1+x)^{n}$.

(1) Find the coefficient of the term containing $x^n$ in the expansion of $(1+x)^{2n-1}$, and simplify the expression $C_{n-1}^{0}C_{n}^{n} + C_{n-1}^{1}C_{n}^{n-1} + \cdots + C_{n-1}^{n-1}C_{n}^{1}$;

(2) Simplify the expression $(C_{n}^{1})^{2} + 2(C_{n}^{2})^{2} + \cdots + n(C_{n}^{n})^{2}$.

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