
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 $a^2 + b^2 + c^2 + d^2 + e^2 = 100.$  Let $N$ be the maximum value of
\[ac + 3bc + 4cd + 8ce,\]and let $a_N,$ $b_N$, $c_N,$ $d_N,$ $e_N$ be the values of $a,$ $b,$ $c,$ $d,$ $e,$ 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}
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