
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
Find the number of combinations of $k$ numbers $\{j_{1}, j_{2}, \cdots, j_{k}\}$ drawn from the set $\{1,2, \cdots, n\}$ that satisfy the following conditions:
1. $1 \le j_{1} < j_{2} < \cdots < j_{k} \le n$
2. $j_{h+1} - j_{h} \ge m$, for $h = 1,2, \cdots, k-1$, where $m > 1$ is a fixed positive integer
3. There exists some $h_{0}$, $1 \le h_{0} \le k-1$, such that $j_{h_{0}+1} - j_{h_{0}} \ge m+1$

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