
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]


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We call a polynomial of the form $ax^{2}+bx+c\left(a\neq 0\right)$ a quadratic trinomial in terms of $x$. Based on the method of factoring quadratic trinomials of the form $x^{2}+\left(p+q\right)x+pq$, we now introduce a method to factorize general quadratic trinomials within the real number range using the method of finding the roots of a quadratic equation. For $ax^{2}+bx+c=a(x^{2}+\frac{b}{a}x+\frac{c}{a})=a[x^{2}-(x_{1}+x_{2})x+x_{1}x_{2}]=a(x-x_{1})(x-x_{2})$, this means that when factoring a quadratic trinomial $ax^{2}+bx+c\left(a\neq 0\right)$, we can first find the roots $x_{1}$ and $x_{2}$ of the equation $ax^{2}+bx+c=0$, and then write it as $ax^{2}+bx+c=a(x-x_{1})(x-x_{2})$. Example: Factorize within the real number range: $3x^{2}-x-1$.

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