
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
On the interval \(\left[\frac{1}{2}, 2\right]\), the functions \(f(x) = x^{2} + p x + q\) and \(g(x) = 2 x + \frac{1}{x^{2}}\) achieve the same minimum value at the same point. Determine the maximum value of \(f(x)\) on \(\left[\frac{1}{2}, 2\right]\).

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