
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
Read and think

Read the following example: Solve the inequality: $|2x| \gt 1$. Solution: ① When $2x \gt 0$, that is, $x \gt 0$, the original inequality can be transformed into a linear inequality $2x \gt 1$. Solve this inequality, we get $x \gt \frac{1}{2}$. Therefore, $x \gt \frac{1}{2}$. ② When $2x \lt 0$, that is, $x \lt 0$, the original inequality can be transformed into a linear inequality $-2x \gt 1$. Solve this inequality, we get $x \lt -\frac{1}{2}$, (according to $)$. Therefore, $x \lt -\frac{1}{2}$. ③ When $2x=0$, that is, $x=0$, the original inequality can be transformed into $0 \gt 1$, which is not true, so the inequality has no solution. Therefore, the solution to the inequality is $x \lt -\frac{1}{2}$ or $x \gt \frac{1}{2}$.

Task:

$(1)$ Fill in the blank: In the above solution process, "according to" refers to ______;

$(2)$ Following the example, use the method of classification to solve the inequality: $|2x+1| \gt 3$.

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