
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
Let \( I \) be a subinterval of \( (0, +\infty) \), and \( f(x) \) be a non-negative function on \( I \).

For any \( x_{1}, x_{2} \in I \), if \( f\left(\sqrt{x_{1} \cdot x_{2}}\right) \leq \sqrt{f\left(x_{1}\right) \cdot f\left(x_{2}\right)} \) always holds, then \( f(x) \) is called a geometrically concave function on \( I \). If \( f\left(\sqrt{x_{1} \cdot x_{2}}\right) \geq \sqrt{f\left(x_{1}\right) \cdot f\left(x_{2}\right)} \) always holds, then \( f(x) \) is called a geometrically convex function on \( I \).

Given the function \( f(x) = \left(\frac{1}{3}\right)^{x} - \left(\frac{1}{4}\right)^{x} \) for \( x \in [1, +\infty) \), determine whether \( f(x) \) is a geometrically concave function or a geometrically convex function on \( [1, +\infty) \), and provide a proof.

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