
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 inequality system about $x$: $\left\{\begin{array}{c}6-(2x+5)>-15\\ \frac{x+3}{2}-t<x\end{array}\right.$, <br/>$(1)$ If $t=2$, then the smallest integer solution of this inequality system is ______; <br/>$(2)$ If the solution set of this inequality system is $-5 < x < 8$, then the value of $t$ is ______; <br/>$(3)$ If this inequality system has exactly three integer solutions, then the range of $t$ is ______.

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