
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 line integral

$$
\int_{L} \frac{y}{3} d x - 3 x d y + x d z
$$

along the curve \( L \), which is given parametrically by

$$
\begin{cases}
x = 2 \cos t \\
y = 2 \sin t \\
z = 1 - 2 \cos t - 2 \sin t
\end{cases}
\quad \text{for} \quad 0 \leq t \leq \frac{\pi}{2}
$$

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