
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 $f(x)=|x-a|+|x+3a-2}$ and $g(x)=-x^{2}+2ax+1$ where $a\in R$.
$(1)$ When $a=2$, solve the inequality $f(x)\geqslant 7$ with respect to $x$.
$(2)$ If for all $x_{1}$, $x_{2}\in R$, $f(x_{1}) \gt g(x_{2})$ holds, find the range of values for $a$.

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