
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 $f(x) = a^x + b$ satisfy the conditions $f(0) = 2$ and $f(x+1) = 2f(x) - 1$ for any $x \in \mathbb{R}$.  
(1) Find the explicit formula for $f(x)$.  
(2) Let the domain of function $g(x)$ be $[-2, 2]$, and within this domain $g(x) = f(x)$. If the graph of function $h(x)$ is symmetric to the graph of $g(x)$ about the line $y = x$, find $h(x)$.  
(3) Find the range of the function $y = g(x) + h(x)$.

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