
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 \(a^2 + b^2 + c^2 + d^2 + e^2 = 504.\) Determine the maximum value of the expression
\[ac + 3bc + 4cd + 6ce,\]
and find \(a_M, b_M, c_M, d_M, e_M\) that achieve this maximum value. Compute \(M + a_M + b_M + c_M + d_M + e_M.\)

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