
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
In the range $0 \leq r \le n \leq 63$, how many pairs of $(n, r)$ are there for which $C_{n}^{r} = \frac{n!}{r!(n-r)!}$ is an even number?

(Note: This problem is from the Japanese Mathematical Olympiad, 1991.)

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