
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

Determine the value of the determinant

$$
\begin{aligned}
& \left\lvert\, \begin{array}{lllll}
\binom{0}{0} & \binom{1}{1} & \binom{2}{2} & \ldots & \binom{n}{n} \\
\binom{1}{0} & \binom{2}{1} & \binom{3}{2} & \ldots & \binom{n+1}{n}
\end{array}\right. \\
& D=\binom{2}{0} \quad\binom{3}{1} \quad\binom{4}{2} \quad \ldots\binom{n+2}{n} \\
& \binom{n}{0}\binom{n+1}{1}\binom{n+2}{2} \ldots\binom{2 n}{n}
\end{aligned}
$$

Determine the determinant's value. (\(n\) is a positive integer.)

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