
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
Integrate the following equations:
1. \((1+y^{2}) \, dx - (1+x^{2}) \, y \, dy = 0\)
2. \(x y \, y' + x^{2} - 1 = 0\)
3. \(\sec^{2}(x) \tan(y) \, dx + \sec^{2}(y) \tan(x) \, dy = 0\)
4. \(y \, y' = \frac{1 + 2x}{y}\)
5. \(y' = e^{x^{2}} \ln(y)\)
6. \(\frac{dx}{dt} = 2t \sqrt{x}\)

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