
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 that \( f(n) = \left( C_{n}^{2} \right)^{2} \cdots \left( C_{n}^{1} \right)^{2} + \left( C_{n}^{2} \right)^{2} - \left( C_{n}^{3} \right)^{2} + \cdots + (-1)^{k} \left( C_{n}^{k} \right)^{2} + \cdots + (-1)^{n} \left( C_{n}^{n} \right)^{2} \) (where \( n \) is an even number), what is \( f(1990) \)?

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