
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 \( a, b, c, d, e \in \mathbb{R}^+ \). Find the maximum value of \(\frac{a b + 2 b c + 2 c d + d e}{a^2 + 3 b^2 + 3 c^2 + 5 d^2 + e^2}\).

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