
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
Define a function $f(x) = 3^{x-m}$ on the interval $[2,4]$ (where $m$ is a constant) whose graph passes through the point $(2,1)$. Let $f^{-1}(x)$ be the inverse function of $f(x)$. Then, the range of the function $F(x) = [f^{-1}(x)]^2 - f^{-1}(x^2)$ is.

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