
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 $a$ and $b$ are two distinct real numbers, let set $M=\{a^2-4a, -1\}$ and set $N=\{b^2-4b+1, -2\}$. Define a mapping $f: x \rightarrow x$ such that it maps elements from $M$ to $N$, and the image is still $x$. Find the value of $a+b$.

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