
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
Let \( x, y, z, u, v \in \mathbf{R}_{+} \). The maximum value of 
\[ 
f = \frac{x y + y z + z u + u v}{2 x^{2} + y^{2} + 2 z^{2} + u^{2} + 2 v^{2}} 
\] 
is $\qquad$ .

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