
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 $f(x)=|2x-a|+8x$, where $x > -2$ and $a > 0$.

$(1)$ When $a=1$, find the solution set of the inequality $f(x) \geqslant 2x+1$;

$(2)$ If the graph of the function $g(x)=f(x)-7x-a^{2}+3$ falls within the region $\begin{cases} & x > -2 \\ & y\geqslant 0 \end{cases}$, find the range of the real number $a$.

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