
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
For \( m \in \mathbf{N} \) and \( n \in \mathbf{N}^{*} \), the binomial coefficient is defined as \( C_{n}^{m} \). Additionally, the "binomial theorem" states: 
\[ (a+b)^{n}=a^{n}+C_{n}^{1} a^{n-1} b+\cdots+C_{n}^{i} a^{n-i} b^{i}+\cdots+b^{n} \]
Using this, find:
\[ 9^{n} \cdot C_{2 n}^{2 n}+9^{n-1} \cdot C_{2 n}^{2 n-2}+9^{n-2} \cdot C_{2 n}^{2 n-4}+\cdots+9^{2} \cdot C_{2 n}^{4}+9 \cdot C_{2 n}^{2}+1 \]
Express the result in the simplest form in terms of \( n \).

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