
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
Calculate the circulation of the vector field given in cylindrical coordinates: \( \vec{A} = \rho \sin \varphi \, \mathrm{e}_{\rho} + \rho z \, \mathrm{e}_{\varphi} + \rho^{3} \, \mathrm{e}_{z} \), along the curve \( L: \{\rho = \sin \varphi, z = 0, 0 \leqslant \varphi \leqslant \pi\} \) directly and using Stokes' theorem.

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