
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

$$
\vec{a}=\left(y^{2}+z^{2}\right) \vec{i}+\left(x y+y^{2}\right) \vec{j}+(x z+z) \vec{k}
$$

through the closed surface $\Sigma$, which is the complete surface of a cylinder

$$
x^{2}+y^{2}=1, \quad z=0, \quad z=1
$$

(with an outward normal).

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