
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 $x,$ $y,$ $z,$ $v,$ $w$ be positive real numbers such that $x^2 + y^2 + z^2 + v^2 + w^2 = 2025.$ Let $N$ be the maximum value of
\[ xz + 3yz + 5zv + 2zw, \]
and let $x_N,$ $y_N$, $z_N,$ $v_N,$ $w_N$ be the values of $x,$ $y,$ $z,$ $v,$ $w,$ respectively, that produce the maximum value of $N.$ Find $N + x_N + y_N + z_N + v_N + w_N.$

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