
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
For any real number $a$, we use $\left[a\right]$ to represent the largest integer not greater than $a$, then $a-1 \lt \left[a\right]\leqslant a$, for example: $\left[1.6\right]=1$, $\left[2023\right]=2023$, $\left[-3.07\right]=-4$. Please answer the following questions based on the above information.<br/>$(1)$ Fill in the blank: $[{\sqrt{17}}]=$____;<br/>$(2)$ If $[{x+2}]=2x-\frac{1}{4}$, then the value of $x$ is ____.

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