
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 that the square root of $a+1$ is $\pm 5$, the cube root of $b$ is $-2$, and $c$ is the integer part of $\sqrt{12}$;  
$(1)$ Write down the values of $a$, $b$, and $c$ directly;  
$(2)$ If $x$ is the decimal part of $\sqrt{12}$, find the arithmetic square root of $(\sqrt{12}+3)-x$.

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