
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
Given the function $F(x) = \begin{cases} f(x), & f(x) \geq g(x) \\ g(x), & f(x) < g(x) \end{cases}$, where $f(x) = \log_{2}(x^{2}+1)$ and $g(x) = \log_{2}(|x|+7)$.

1. Write the analytical expression of function $F(x)$ in piecewise form on the set of real numbers $\mathbb{R}$.
2. Find the minimum value of function $F(x)$.

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