
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
(1) The derivative of the function $y=\frac{x^2}{\sin x}$ is ______________.

(2) For the function $y=x^3+ax^2+bx+a^2$, if there is an extremum of $10$ at $x=1$, then the values of $a$ and $b$ are respectively ______________.

(3) When proving $(n+1)(n+2)\ldots(n+n)=2^n\cdot1\cdot3\cdot\ldots\cdot(2n-1)$ $(n\in\mathbb{N})$ using mathematical induction, the algebraic expression that needs to be added to the left side when going from "$k$" to "$k+1$" is ______________.

(4) Let $x^3+ax+b=0$, where $a$ and $b$ are real numbers. Among the following conditions, the ones that ensure the cubic equation has only one real root are ______________ (write down all correct condition numbers).

    (1) $a=-3, b=-3$; (2) $a=-3, b=2$; (3) $a=-3, b > 2$; (4) $a=0, b=2$;

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