
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 $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. Consider the system of equations where the first equation is \( 3x^2 - 45\lfloor x \rfloor + 60 = 0 \) and the second equation is \( 2x - 3\lfloor x \rfloor + 1 = 0 \). Determine the number of real solutions of the system.

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