
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
If the real numbers \( x > y > z > t > 0 \), to ensure that \( \log_{\frac{x}{y}} 2009 + \log_{\frac{x}{z}} 2009 + \log_{\frac{x}{4}} 2009 \geq k \log_{\frac{x}{4}} 2009 \) always holds, the maximum value of \( k \) is \(\quad\).

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