
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
Apply the factorial binomial theorem to find the values of the following sums:

a) \( C_{n}^{0} C_{m}^{k} + C_{n}^{1} C_{m}^{k-1} + C_{n}^{2} C_{m}^{k-2} + \ldots + C_{n}^{k} C_{m}^{0} \);

b) \( C_{m}^{0} C_{n}^{k} - C_{m+1}^{1} C_{n}^{k-1} + C_{m+2}^{2} C_{n}^{k-2} - \cdots + (-1)^{k} C_{m+k}^{k} C_{n}^{0} \).

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