
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 minimum value of  $k$  such that there exists two sequence  ${a_i},{b_i}$  for  $i=1,2,\cdots ,k$  that satisfies the following conditions.

(i) For all  $i=1,2,\cdots ,k,$   $a_i,b_i$  is the element of  $S=\{1996^n|n=0,1,2,\cdots\}.$ (ii) For all  $i=1,2,\cdots, k, a_i\ne b_i.$ (iii) For all  $i=1,2,\cdots, k, a_i\le a_{i+1}$  and  $b_i\le b_{i+1}.$ (iv)  $\sum_{i=1}^{k} a_i=\sum_{i=1}^{k} b_i.$ 

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