
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
Given real numbers $a\geqslant 0$, $b\geqslant 0$, and $a+b=4$, let $w=a^{2}+ab+b^{2}$. Let $w_{1}$ and $w_{2}$ be the maximum and minimum values of the algebraic expression $w$, respectively. Find the value of $w_{1}-w_{2}$.

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