
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
Observation: Because $\sqrt{4} \lt \sqrt{5} \lt \sqrt{9}$, that is, $2 \lt \sqrt{5} \lt 3$, so the integer part of $\sqrt{5}$ is $2$, and the decimal part is $\sqrt{5}-2$. Please solve the following problem after observing the above pattern:  
$(1)$ Define the symbol $\left[m\right]$ to represent the integer part of a real number $m$, for example: $[\frac{2}{3}]=0$, $[\sqrt{6}]=2$. According to this rule, what is the value of $[\sqrt{10}+1]$?  
$(2)$ If the integer part of $\sqrt{11}$ is $a$, the decimal part is $b$, and $|c|=\sqrt{11}$, find the value of $c\left(a-b-6\right)+12$.

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