
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 $a$ and $b$ be real numbers, and let the sets $M = \{-1, \frac{b}{a}, 1\}$ and $N = \{a, b, b-a\}$. If the function $f: x \to x$ maps elements from set $M$ to set $N$ such that each element $x$ in $M$ maps to the element $x$ in $N$ (i.e., the identity mapping), calculate the value of $a+b$.

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