
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 $f(x)=\frac{a}{x-1}+ax(a > 0)$ has a minimum value of $15$ on the interval $(1,+\infty)$, and the function $g(x)=|x+a|+|x+1|$.

$(1)$ Find the value of the real number $a$;

$(2)$ Find the minimum value of the function $y=g(x)$.

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