
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 sum of all integer solutions of the inequality system about $x$ $\left\{{\begin{array}{l}{2x+1>x+a}\\{\frac{x}{2}+1\geq\frac{5}{2}x-9}\end{array}}\right.$ is $14$, then the value of the integer $a$ is ______.

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