
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
Define the length of intervals $(m, n)$, $[m, n]$, $(m, n]$, and $[m, n)$ as $n-m$, where $n>m$.

(1) If the solution set of the inequality $ax^2+12x-3>0$ forms an interval with a length of $2\sqrt{3}$, find the value of the real number $a$;  
(2) Find the range of the length of the interval formed by the solution set of the inequality $x^2-3x+(\sin\theta+\cos\theta)<0$ ($\theta\in\mathbb{R}$);  
(3) Given that the sum of the lengths of the intervals formed by the solution set of the system of inequalities $$\begin{cases} \frac{7}{x+2}>1 \\ \log_{2}x+\log_{2}(tx+2t)<3\end{cases}$$ is 5, find the range of the real number $t$.

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