
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
1. For all natural numbers \( n \), we have the equality:

$$
\sum_{k=0}^{n}\binom{n}{k}^{2}=\binom{2 n}{n}
$$

2. For all natural numbers \( a \), \( N \), and \( k \), we have:

$$
\binom{N+a+1}{k+1}-\binom{N}{k+1}=\sum_{j=0}^{a}\binom{N+j}{k}
$$

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