
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 non-negative real numbers \( x, y, z, w \) that satisfy \( x^{2}+y^{2}+z^{2}+w^{2}+x+2y+3z+4w=\frac{17}{2} \), find the maximum and minimum values of \( x + y + z + w \).

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