
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
Define $a * b$ as the "vector product" of vectors $a$ and $b$, with the magnitude $|a * b| = |a| \cdot |b| \cdot \sin \theta$, where $\theta$ is the angle between vectors $a$ and $b$. If $u = (2, 0)$, and $u - v = (1, -\sqrt{3})$, then $|u * (u + v)| =$ _____________

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