
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
For two functions $f(x)$ and $g(x)$ with the same domain, if there exist real numbers $m$ and $n$ such that $h(x) = mf(x) + ng(x)$, then the function $h(x)$ is said to be generated by the "base functions $f(x), g(x)$".
(Ⅰ) If $h(x) = 2x^2 + 3x + 1$ is generated by the functions $f(x) = x^2 + ax$ and $g(x) = x + b$, with $b \in \left[ \frac{1}{2}, 1 \right]$, find the range of values for $a + 2b$;
(Ⅱ) Try to generate a function $h(x)$ using the "base functions $f(x) = \log_{4}(4^x + 1), g(x) = x - 1$" that satisfies the following conditions:
① It is an even function;
② It has a minimum value of 1.
Find the expression for $h(x)$.

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