
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 $f(x)$ satisfies $f(2^x) = x + 1$.
1. Find the explicit formula for the function $f(x)$.
2. Find the minimum value of the function $y = [f(x)]^2 + f(2x)$.
3. Let the function $g(x)$ be the inverse function of $y = f(x) - 1$, and let $h(x) = f(x) + g(x)$. If the equation $h(x) - a = 0$ has a root in the interval $(1, 2)$, find the range of the real number $a$.

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