
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 $(x+1)(2x-3)=2x^2-x-3$, the quotient of the polynomial $2x^2-x-3$ divided by $x+1$ is $2x-3$.

Solve the following:
(1) Given a quadratic polynomial in $x$ is divided by $x-5$, with a quotient of $2x+6$ and a remainder of $2$, find this polynomial.
(2) Given a polynomial in $x$ of $3x^2+mnx+n$ is divided by $x+1$ with a quotient of $3x-5$ and a remainder of $x$, find the values of $m$ and $n$.
(3) Given a cubic polynomial in $x$ has a remainder of $2x-5$ when divided by $x^2-1$, and a remainder of $-3x+4$ when divided by $x^2-4$, find this cubic polynomial.

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