
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 $f(x)=ax^{2}+1(a > 0)$ and $g(x)=x^{3}+bx$.

$(1)$ If the curve $y=f(x)$ and the curve $y=g(x)$ have a common tangent line at their intersection point $(1,c)$, find the values of $a$ and $b$;

$(2)$ When $a^{2}=4b$, find the monotonic intervals of the function $f(x)+g(x)$, and find its maximum value on the interval $(-\infty,-1]$.

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