
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
For which values of \( c \) do the following equations have solutions?

a) \( x + \left\lfloor \frac{x}{2} \right\rfloor = c \)

b) \( x - \left\lfloor \frac{x}{2} \right\rfloor = c \)

If solutions exist, how many are there, and what are the solutions?

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