
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
Verify that the given function is a solution to the given differential equation:
1) \( y=\sqrt{x}, \ 2 y y^{\prime}=1 \)
2) \( \ln x \ln y=c, \ y \ln y \, dx + x \ln x \, dy = 0 \)
3) \( s=-t-\frac{1}{2} \sin 2t, \ \frac{d^{2} s}{d t^{2}} + \tan t \frac{d s}{d t} = \sin 2t \)

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