
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  $r\in\mathbb{R}$ . Alice and Bob plays the following game:
An equation with blank is written on the blackboard as below:  $$ S=|\Box-\Box|+|\Box-\Box|+|\Box-\Box| $$  Every round, Alice choose a real number from  $[0,1]$  (not necessary to be different from the numbers chosen before)  and Bob fill it in an empty box. After 6 rounds, every blank is filled and  $S$  is determined at the same time. If  $S\ge r$  then Alice wins, otherwise Bob wins.
Find all  $r$  such that Alice can guarantee her victory.

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