
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
Find the flux of the vector field \( \mathbf{a} = \mathbf{i} - \mathbf{j} + xyz \mathbf{k} \) through the circle \( S \), obtained by the intersection of the sphere \( x^2 + y^2 + z^2 \leq R^2 \) with the plane \( y = x \). Take the side of the circle facing the positive part of the \( O_x \)-axis.

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