
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 \( x \) be a real number. Define \( \lceil x \rceil \) as the smallest integer not less than \( x \). For example, \( \lceil \pi \rceil = 4 \) and \( \lceil -\pi \rceil = -3 \). The sum of all real roots of the equation \( \lceil 3x + 1 \rceil = 2x - \frac{1}{2} \) is equal to ________ .

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