
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
Define the ☆ operation. Observe the following operations:

| $(+3)$☆$(+15)=+18$ | $(-14)$☆$(-7)=+21$ |
|---------------------|-------------------|
| $(-2)$☆$(+14)=-16$ | $(+15)$☆$(-8)=-23$ |
| $0$☆$(-15)=+15$    | $(+13)$☆$0=+13$   |

$(1)$ Please carefully consider the above operations and summarize the rules of the ☆ operation: When two numbers are operated with ☆, if they have the same sign, the result is positive and the absolute values are added; if they have different signs, the result is negative and the absolute values are added. Specifically, operating with $0$ and any number, or any number and $0$, results in the absolute value of that number.

$(2)$ Calculate: $\left(+11\right)$☆$[0$☆$(-12)]=\_\_\_\_\_\_$.

$(3)$ If $2\times (2$☆$a)-1=3a$, find the value of $a$.

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