
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 $x^2 + y^2 + z^2 + v^2 + w^2 = 2024.$ Let $L$ be the maximum value of
\[ xz + 3yz + 3zv + 3zw, \]
and let $x_L,$ $y_L,$ $z_L,$ $v_L,$ $w_L$ be the values of $x,$ $y,$ $z,$ $v,$ $w,$ that produce the maximum value of $L.$ Find $L + x_L + y_L + z_L + v_L + w_L.$

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