
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 the functions $f_1(x)$, $f_2(x)$, and $h(x)$, if there exist real numbers $a$ and $b$ such that $h(x) = a \cdot f_1(x) + b \cdot f_2(x)$, then $h(x)$ is called the generating function of $f_1(x)$ and $f_2(x)$.
(Ⅰ) Given two sets of functions below, is $h(x)$ the generating function of $f_1(x)$ and $f_2(x)$? Explain your reasoning;
   Set 1: $f_{1}(x) = \sin x,  f_{2}(x) = \cos x,  h(x) = \sin\left(x + \frac{\pi}{3}\right)$;
   Set 2: $f_1(x) = x^2 - x, f_2(x) = x^2 + x + 1, h(x) = x^2 - x + 1$;
(Ⅱ) Let $f_{1}(x) = \log_{2}x,  f_{2}(x) = \log_{\frac{1}{2}}x,  a = 2,  b = 1$, find the generating function $h(x)$. If the inequality $3h^2(x) + 2h(x) + t < 0$ has a solution within $x \in [2, 4]$, find the range of the real number $t$;
(Ⅲ) Let $f_{1}(x) = x,  f_{2}(x) = \frac{1}{x}  (1 \leq x \leq 10)$, take $a = 1, b > 0$, find the generating function $h(x)$ such that $h(x) \geq b$ always holds, find the range of $b$.

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