
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 $f(x)=ax^3+x$ and $g(x)=x^2+px+q$.

$(1)$ If the function $f(x)$ has an extremum at $x=1$, find the value of the real number $a$;

$(2)$ When $p=4$ and $q=3$, if for any $x \geqslant 1$, $g(x) \geqslant (6+\lambda)x-\lambda\ln x+3$ always holds, find the range of the real number $\lambda$.

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