
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
Solve the equations:
1) \( y' - y \cot(x) = \sin(x) \)
2) \( x^2 y^2 y' + x y^3 = 1 \)
3) \( y \, dx - (3x + 1 + \ln(y)) \, dy = 0 \) with the initial condition \( y\left(-\frac{1}{3}\right) = 1 \).

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