
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
Initially given  $31$  tuplets  $$ (1,0,0,\dots,0),(0,1,0,\dots,0),\dots, (0,0,0,\dots,1) $$    
were written on the blackboard. At every move we choose two written  $31$  tuplets as  $(a_1,a_2,a_3,\dots, a_{31})$  and  $(b_1,b_2,b_3,\dots,b_{31})$ , then write the  $31$  tuplet  $(a_1+b_1,a_2+b_2,a_3+b_3,\dots, a_{31}+b_{31})$  to the blackboard too. Find the least possible value of the moves such that one can write the  $31$  tuplets  $$ (0,1,1,\dots,1),(1,0,1,\dots,1),\dots, (1,1,1,\dots,0) $$    
to the blackboard by using those moves.

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