
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 \( x^2 + y^2 + z^2 + v^2 + w^2 = 2023 \). Let \( M \) be the maximum value of
\[ xz + 3yz + 4zv + 6zw, \]
and let \( x_M, y_M, z_M, v_M, w_M \) be the values of \( x, y, z, v, w \), respectively, that produce the maximum value of \( M \). Find \( M + x_M + y_M + z_M + v_M + w_M \).

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