
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
How many solutions does the equation $\left[2 x - x^{2}\right] + 2\{\cos 2 \pi x\} = 0$ have? Indicate the smallest and largest among them. Here, $[a]$ is the integer part of the number $a$ (the largest integer not exceeding $a$), and $\{a\} = a - [a]$ is the fractional part of the number $a$.

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