
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
Given a positive integer  $n \ge 2$ . Find all  $n$ -tuples of positive integers  $(a_1,a_2,\ldots,a_n)$ , such that  $1<a_1 \le a_2 \le a_3 \le \cdots \le a_n$ ,  $a_1$  is odd, and
(1)  $M=\frac{1}{2^n}(a_1-1)a_2 a_3 \cdots a_n$  is a positive integer;
(2) One can pick  $n$ -tuples of integers  $(k_{i,1},k_{i,2},\ldots,k_{i,n})$  for  $i=1,2,\ldots,M$  such that for any  $1 \le i_1 <i_2 \le M$ , there exists  $j \in \{1,2,\ldots,n\}$  such that  $k_{i_1,j}-k_{i_2,j} \not\equiv 0, \pm 1 \pmod{a_j}$ .

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