
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
If the integer $a$ makes the system of inequalities about $x$ $\left\{\begin{array}{l}{4x-a≥1}\\{\frac{x+13}{2}≥x+2}\end{array}\right.$ have exactly $6$ integer solutions, then the sum of all values of the integer $a$ that satisfy the condition is ______.

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