
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
Let $[a]$ denote the greatest integer less than or equal to $a$. For example, $[\sqrt{2}] = 1$ and $[-\sqrt{2}] = -2$. Find the sum of all solutions to the equation
\[ 
[3x + 1] = 2x - \frac{1}{2} 
\]
(China Junior High School Mathematics League, 1987).

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