
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
Let  $T$  be the set of ordered triples  $(x,y,z)$ , where  $x,y,z$  are integers with  $0\leq x,y,z\leq9$ . Players  $A$  and  $B$  play the following guessing game. Player  $A$  chooses a triple  $(x,y,z)$  in  $T$ , and Player  $B$  has to discover  $A$ *'s* triple in as few moves as possible.  A *move* consists of the following:  $B$  gives  $A$  a triple  $(a,b,c)$  in  $T$ , and  $A$  replies by giving  $B$  the number  $\left|x+y-a-b\right |+\left|y+z-b-c\right|+\left|z+x-c-a\right|$ . Find the minimum number of moves that  $B$  needs to be sure of determining  $A$ *'s*  triple.

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