
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
Definition: For a rational number $x$, we call $\{x\}$ the associated number of $x$; if $x \geqslant 0$, then $\{x\} = 2x - 1$; if $x \lt 0$, then $\{x\} = -2x + 1$. For example: $\{1\} = 2 \times 1 - 1 = 1$.

$(1)$ $\{\frac{3}{2}\} = \_\_\_\_\_\_$; $\{-1\} = \_\_\_\_\_\_$;

$(2)$ Given $a \gt 0$, $b \lt 0$, and $\{a\} = \{b\}$, find the value of the expression $\left(a+b\right)^{2} - 2a - 2b$.

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