
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
Find the maximum value $M$ that the expression $7x + 10y + z$ can take when $x, y, z$ are real numbers satisfying $x^{2} + 2x + \frac{1}{5}y^{2} + 7z^{2} = 6$. In which cases is this value attained?

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