
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
Given the mapping $f: x \to y = -x^2 + 2x$, where $M$ and $N$ are equal to the set of real numbers, determine the range of values for $P$ in $N$ such that there does not exist a preimage in $M$.

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