
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 real number $r$, a game is played by two players, A and B, as follows.

An expression with three absolute values is written on the blackboard: $S = |\square - \square| + |\square - \square| + |\square - \square|$, where the 6 boxes " $\square$ " are yet to be filled with numbers. Each round, A selects a real number from the interval $[0,1]$ (the same number can be selected in different rounds), and B places that number in one of the empty slots. After six rounds, all 6 slots are filled, thus determining the value of $S$. If $S \geqslant r$, A wins; otherwise, B wins. Find all real numbers $r$ such that A has a winning strategy.

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