
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
Determine the smallest real number  $C$  such that the inequality \[ C(x_1^{2005} +x_2^{2005} + \cdots + x_5^{2005}) \geq x_1x_2x_3x_4x_5(x_1^{125} + x_2^{125}+ \cdots + x_5^{125})^{16}  \] holds for all positive real numbers  $x_1,x_2,x_3,x_4,x_5$ .

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