
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  $n$ , there are  $n$  boxes  $B_1,...,B_n$ . The following procedure can be used to add balls.  $$ \text{(Procedure) Chosen two positive integers }n\geq i\geq j\geq 1\text{, we add one ball each to the boxes }B_k\text{ that }i\geq k\geq j. $$  For positive integers  $x_1,...,x_n$  let  $f(x_1,...,x_n)$  be the minimum amount of procedures to get all boxes have its amount of balls to be a multiple of 3, starting with  $x_i$  balls for  $B_i(i=1,...,n)$ . Find the largest possible value of  $f(x_1,...,x_n)$ . (If  $x_1,...,x_n$  are all multiples of 3,  $f(x_1,...,x_n)=0$ .)

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