
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
Compute the circulation of the vector field $\mathbf{a} = y e^{x y} \mathbf{i} + x e^{x y} \mathbf{j} + x y z \mathbf{k}$ along the path $L$, obtained by the intersection of the cone $x^{2} + y^{2} = (z-1)^{2}$ with the coordinate planes.

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