
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
There exist positive integers $x$, $y$, and $z$ such that
\[4 \sqrt{\sqrt[3]{7} - \sqrt[3]{6}} = \sqrt[3]{x} + \sqrt[3]{y} - \sqrt[3]{z}.\]
Find $x + y + z$.

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