
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
By using the formula method, we can transform some polynomials of the form $ax^{2}+bx+c\left(a\neq 0\right)$ into the form $a\left(x+m\right)^{2}+n$. We call this transformation method the method of matching polynomials $ax^{2}+bx+c\left(a\neq 0\right)$. By using the method of matching polynomials and the square difference formula, we can factorize some polynomials. For example, ${x}^{2}+4x-5={x}^{2}+4x+{(\frac{4}{2})}^{2}-{(\frac{4}{2})}^{2}-5={(x+2)}^{2}-9=(x+2+3)(x+2-3)=(x+5)(x-1)$. Based on the above material, answer the following questions.<br/>$(1)$ Factorize the polynomial: $x^{2}+2x-3$;<br/>$(2)$ Find the minimum value of the polynomial $x^{2}+4x+5$.

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