
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

Write the expression

$$
K=\frac{\frac{1}{a+b}-\frac{2}{b+c}+\frac{1}{c+a}}{\frac{1}{b-a}-\frac{2}{b+c}+\frac{1}{c-a}}+\frac{\frac{1}{b+c}-\frac{2}{c+a}+\frac{1}{a+b}}{\frac{1}{c-b}-\frac{2}{c+a}+\frac{1}{a-b}}+\frac{\frac{1}{c+a}-\frac{2}{a+b}+\frac{1}{b+c}}{\frac{1}{a-c}-\frac{2}{a+b}+\frac{1}{b-c}}
$$

in a simpler form. Calculate its value if \( a=5, b=7, c=9 \). Determine the number of operations (i.e., the total number of additions, subtractions, multiplications, and divisions) required to compute \( K \) from the simplified expression and from the original form. Also, examine the case when \( a=5, b=7, c=1 \). What are the benefits observed from algebraic simplifications in this context?

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