
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 area of a parallelogram constructed on vectors \(\vec{a}\) and \(\vec{b}\), given the lengths of its diagonals, \(\cos (\widehat{\vec{a}, \vec{b}})\), and \(\| \vec{a} \times \vec{b} \|\), where

\[
\vec{a} = 5 \vec{p} + 2 \vec{q}, \quad \vec{b} = \vec{p} - 3 \vec{q}, \quad |\vec{p}| = 4 \sqrt{2}, \quad |\vec{q}| = 6, \quad (\widehat{\vec{p}, \vec{q}}) = \frac{\pi}{4}
\]

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