
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
Observe the following equations:  
(1) $C_{2}^{1} = C_{1}C_{1}^{1} + C_{1}^{1}C_{1}$;  
(2) $C_{4}^{2} = C_{2}C_{2}^{2} + C_{2}^{1}C_{2}^{1} + C_{2}^{2}C_{2}$;  
(3) $C_{6}^{3} = C_{3}C_{3}^{3} + C_{3}^{1}C_{3}^{2} + C_{3}^{2}C_{3}^{1} + C_{3}^{3}C_{3}$,  
By summarizing their characteristics, we can make a conjecture:  
 $C_{2n}^{n} =$    ．

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