
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

Solve the equation

$$
\frac{x+7}{x-8}-6=\frac{5 x-55}{7-x}
$$

and determine the validity of the following statements:

1. If the numerators are equal, then the denominators must also be equal, which leads to the absurd conclusion that \(8=7\).
2. The equation implies \(8-x=7-x\), which is a contradiction, so there is no solution for \(x\).
3. The equation \(5x-55=5x-55\) is an identity, implying \(0=0\), so the equation holds for all values of \(x\).
4. There are specific values \(x=8\) and \(x=7\) that must be excluded from the solutions.

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