
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
The sum of positive numbers \(a, b, c,\) and \(d\) does not exceed 4. Find the maximum value of the expression

\[ 
\sqrt[4]{a(b+2c)} + \sqrt[4]{b(c+2d)} + \sqrt[4]{c(d+2a)} + \sqrt[4]{d(a+2b)} 
\]

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