
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 a linear combination of $f_1(x)$ and $f_2(x)$.
(I) Below are two sets of functions, is $h(x)$ a linear combination of $f_1(x)$ and $f_2(x)$ for each set? Please explain your reasoning;
First set: $f_1(x) = \sin x$, $f_2(x) = \cos x$, $h(x) = \sin\left(x + \frac{\pi}{3}\right)$;
Second set: $f_1(x) = x^2 - x$, $f_2(x) = x^2 + x + 1$, $h(x) = x^2 - x + 1$;
(II) Let $f_1(x) = \log_2 x$, $f_2(x) = \log_{\frac{1}{2}} x$, $a = 2$, $b = 1$, find the linear combination $h(x)$. If the inequality $3h^2(x) + 2h(x) + t < 0$ has a solution when $x \in [2, 4]$, find the range of the real number $t$.

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