
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 \(a\), \(b\), \(c\), \(d\), and \(e\) be positive real numbers such that \(a^2 + b^2 + c^2 + d^2 + e^2 = 1000.\) Let \(N\) be the maximum value of 
\[
ac + 3bc + 4cd + 6ce,
\]
and let \(a_N\), \(b_N\), \(c_N\), \(d_N\), and \(e_N\) be the values of \(a\), \(b\), \(c\), \(d\), and \(e\), respectively, that produce the maximum value of \(N\). Find \(N + a_N + b_N + c_N + d_N + e_N.\)

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