
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
Demonstrate - based on a combinatorial interpretation - that

$$
\begin{gathered}
\binom{2 n+1}{2 k+1}=\binom{k}{k}\binom{2 n-k}{k}+\binom{k+1}{k}\binom{2 n-k-1}{k}+\ldots \\
+\binom{m}{k}\binom{2 n-m}{k}+\ldots+\binom{2 n-k}{k}\binom{k}{k}
\end{gathered}
$$


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