
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
Determine all positive integers \( a \) for which the equation

$$
\left(1+\frac{1}{x}\right) \cdot \left(1+\frac{1}{x+1}\right) \cdots \left(1+\frac{1}{x+a}\right) = a - x
$$

has at least one integer solution for \( x \).

For each such \( a \), provide the corresponding solutions.

#### Answer:
Only for \( a = 7 \) are there integer solutions, namely \( x = 2 \) and \( x = 4 \).

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