
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 \( a, b, c \geqslant 1 \), and the positive real numbers \( x, y, z \) satisfy
\[ 
\begin{cases}
a^x + b^y + c^z = 4, \\
x a^x + y b^y + z c^z = 6, \\
x^2 a^x + y^2 b^y + z^2 c^z = 9.
\end{cases}
\]
Then the maximum possible value of \( c \) is \_\_\_\_\_

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