
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 the smallest positive value of \( a \) for which the equation

$$
\frac{\frac{x-a}{2}+\frac{x-2a}{3}}{\frac{x+4a}{5}-\frac{x+3a}{4}}=\frac{\frac{x-3a}{4}+\frac{x-4a}{5}}{\frac{x+2a}{3}-\frac{x+a}{2}}
$$

has an integer root.

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