
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
If \( x \) is a real number and \( k \) is a nonnegative integer, recall that the binomial coefficient \( \binom{x}{k} \) is defined by the formula
\[
  \binom{x}{k} = \frac{x(x - 1)(x - 2) \ldots (x - k + 1)}{k!} \, .
\]
Compute the value of
\[
  \frac{\binom{1/2}{2021} \cdot 4^{2021}}{\binom{4042}{2021}} \, .
\]

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