
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 largest constant \( C \) such that
$$
\sum_{i=1}^{4}\left(x_{i}+\frac{1}{x_{i}}\right)^{3} \geq C
$$
for all positive real numbers \( x_{1}, \cdots, x_{4} \) such that
$$
x_{1}^{3}+x_{3}^{3}+3 x_{1} x_{3}=x_{2}+x_{4}=1.
$$

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