
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
For a positive integer  $n$ ,  $S_n$  is the set of positive integer  $n$ -tuples  $(a_1,a_2, \cdots ,a_n)$  which satisfies the following.

(i).  $a_1=1$ .

(ii).  $a_{i+1} \le a_i+1$ .

For  $k \le n$ , define  $N_k$  as the number of  $n$ -tuples  $(a_1, a_2, \cdots a_n) \in S_n$  such that  $a_k=1, a_{k+1}=2$ .

Find the sum  $N_1 + N_2+ \cdots N_{k-1}$ .

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