
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 function $f(x)=|2x-1|+|x+2|$, $g(x)=|x+1|-|x-a|+a$

$(1)$ Solve the inequality $f(x) > 3$;

$(2)$ For $\forall x_1, x_2 \in \mathbb{R}$, such that $f(x_1) \geqslant g(x_2)$ holds, find the range of values for $a$.

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