
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 $f: A \rightarrow B$, where $A = B = \mathbb{R}$, and the corresponding rule is $f: x \rightarrow y = x^2 + 2x + 3$, if the real number $k \in B$ does not have a pre-image in set $A$, then determine the range of values for $k$.

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