
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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$(1)$ [Reading comprehension] When learning about factorization, we have learned about the method of factoring out the common factor and using formulas (difference of squares formula and perfect square formula). In fact, besides these two methods, there are other methods that can be used for factorization, such as the method of grouping. For example, to factorize $x^{2}+2x-3$, we find that we cannot use the method of factoring out the common factor or directly apply the formulas. In this case, we can use the following method: $x^{2}+2x-3=x^{2}+2\times x\times 1+1^{2}-1-3=\left(x+1\right)^{2}-2^{2}$; <br/> The above solution uses the method of transformation, which transforms the original problem into one that can be further factorized using the difference of squares formula. This method is called the method of grouping. Obviously, the factorization above is not complete. Please complete the factorization of $x^{2}+2x-3$:<br/>$(2)$ [Practical exercise] Factorize $x^{2}+8x+7$ using the method of grouping;<br/>$(3)$ [Extension and innovation] Please explain why, regardless of the value of $x$, the value of the polynomial $-{x^2}+\frac{2}{3}x+1$ is less than $\frac{4}{3}$.

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