
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 \(a,\) \(b,\) \(c,\) \(d,\) \(e\) be positive real numbers such that \(a^2 + b^2 + c^2 + d^2 + e^2 = 504.\) Find the maximum value of
\[ac + 3bc + 4cd + 8ce,\]
and denote this maximum value by \(N.\) Let \(a_N,\) \(b_N,\) \(c_N,\) \(d_N,\) \(e_N\) be the values that achieve this maximum. Compute \(N + a_N + b_N + c_N + d_N + e_N.\)

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