
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
For any positive real numbers  $x_1,\ x_2,\ x_3,\ y_1,\ y_2,\ y_3,\ z_1,\ z_2,\ z_3,$  find the maximum value of real number  $A$  such that if \[ M = (x_1^3+x_2^3+x_3^3+1)(y_1^3+y_2^3+y_3^3+1)(z_1^3+z_2^3+z_3^3+1)  \] and \[ N = A(x_1+y_1+z_1)(x_2+y_2+z_2)(x_3+y_3+z_3),  \] then  $M \geq N$  always holds, then find the condition that the equality holds.

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