
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
Given the field \(\bar{F}=\left(3 y^{2}-5 z^{2}\right) \cdot \bar{i}+\left(5 x^{2}-3 \cdot z^{2}\right) \cdot \bar{j}+8 x y \cdot \bar{k}\) and the pyramid with vertices \(O(0 ; 0 ; 0)\), \(A(3 ; 0 ; 0)\), \(B(0 ; 5 ; 0)\), and \(C(0 ; 0 ; 8)\). Find:

a) the flux of the field \(\bar{F}\) through the face \(ABC\) of the pyramid in the direction of the normal making an acute angle with the \(O z\) axis;

b) the flux of the field \(\bar{F}\) through the outer surface of the pyramid using the Ostrogradsky-Gauss theorem;

c) the circulation of the field \(\bar{F}\) along the closed contour \(ABC\):

\(c_1\)) directly

\(c_2\)) using the Stokes theorem (the contour is traversed in the positive direction relative to the outward normal to the surface of the pyramid).

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