
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  $f:[0,1]\rightarrow [0,1]$  be a continuous and bijective function.
Describe the set:
\[A=\{f(x)-f(y)\mid x,y\in[0,1]\backslash\mathbb{Q}\}\]
<details><summary>Note</summary>You are given the result that *there is no one-to-one function between the irrational numbers and  $\mathbb{Q}$ .*</details>

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