
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
Does there exist a real number \(a\) such that the function \(f(x) = \log_{a}\left(a x^{2} - x\right)\) is an increasing function on the interval \([2, 4]\)? If it exists, find the permissible values of \(a\); if it does not exist, explain the reason.

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