
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 triple integral:

a) \(\iiint_{\Omega}(2 x + 3 y + z) \, dx \, dy \, dz\), where \(\Omega = \{(x, y, z) : x \geq 0, y \geq 0, 0 \leq z \leq 3, x + y \leq 3\}\)

b) \(\iiint_{\Omega}(1 - x)^3 \sqrt{1 - y^2} \, dx \, dy \, dz\), where \(\Omega = \{(x, y, z) : \| x | \leq 1, \| y | \leq 1, \| z | \leq 3\}\)

c) \(\iiint_{\Omega} \frac{x y}{\sqrt{z}} \, dx \, dy \, dz\), where \(\Omega = \left\{(x, y, z) : z^2 \leq x^2 + y^2, x \geq 0, y \geq 0, 0 \leq z \leq 1\right\}\)

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