
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
Elective 4-5: Special Topics on Inequalities

Given the function $f(x)=|2x-a|+a$.  
$(1)$ When $a=2$, solve the inequality $f(x) \leqslant 6$;  
$(2)$ Let the function $g(x)=|2x-1|$, when $x \in \mathbb{R}$, $f(x)+g(x) \geqslant 3$, find the range of values for $a$.

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