
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 $f(x)=|2x-a|+a$, where $a\in \mathbf{R}$, and $g(x)=|2x-1|$.

$(1)$ If when $g(x) \leqslant 5$, it always holds that $f(x) \leqslant 6$, find the maximum value of $a$;

$(2)$ If when $x \in \mathbf{R}$, it always holds that $f(x)+g(x) \geqslant 3$, find the range of values for $a$.

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