
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 $f(x) = ka^x$ (where $k$ is a constant, $a>0$ and $a \neq 1$), the graph of the function passes through point A$(0, 1)$ and point B$(2, 16)$.  
(1) Find the expression of the function;  
(2) $g(x) = b + \frac{1}{f(x) + 1}$ is an odd function, find the value of the constant $b$;  
(3) For any $x_1, x_2 \in \mathbb{R}$ and $x_1 \neq x_2$, compare the size of $f\left( \frac{x_1 + x_2}{2}\right)$ and $\frac{f(x_1) + f(x_2)}{2}$.

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