
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 functions $f(x)=\sqrt{x^2-2x+1}$, $g(x)=-\sqrt{x^2+6x+9}+a$ where $a\in \mathbb{R}$,

$(1)$ Solve the inequality $g(x) > 6$ with respect to $x$;

$(2)$ If the graph of the function $y=2f(x)$ is always above the graph of the function $y=g(x)$, find the range of the real number $a$.

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