
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
Read the following problem-solving process:<br/>Given $(a^{2}+b^{2})^{4}-8(a^{2}+b^{2})^{2}+16=0$, find the value of $a^{2}+b^{2}$.<br/>Xiaoming solved it this way: Let $(a^{2}+b^{2})^{2}=m$, then the original expression can be transformed into $m^{2}-8m+16=0$, which gives $\left(m-4\right)^{2}=0$, so $m=4$.<br/>$\therefore (a^{2}+b^{2})^{2}=4$, hence $a^{2}+b^{2}=\pm 2$.<br/>$(1)$ Is there any mistake in the above solution? If so, please correct it;<br/>$(2)$ Please factorize $\left(a+b\right)^{4}-14\left(a+b\right)^{2}+49$ within the real number range.

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