
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
Which of the following statements are true and which are false?
1) For all numbers \(x\) and \(y\), \(x = 2y\).
2) For any number \(x\), there exists a number \(y\) such that \(x = 2y\).
3) For any number \(y\), there exists a number \(x\) such that \(x = 2y\).
4) There exists a number \(x\) such that for all numbers \(y\), \(x = 2y\).
5) There exists a number \(y\) such that for all numbers \(x\), \(x = 2y\).
6) There exist numbers \(x\) and \(y\) such that \(x = 2y\). (Here and in the next problem \(x\) and \(y\) are real numbers.)

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