
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 the inequality system: $\left\{{\begin{array}{l}{-\frac{1}{3}x>\frac{2}{3}-x,}\\{\frac{1}{2}x-1<\frac{1}{2}(a-2)}\end{array}}\right.$, there are exactly three integer solutions for $x$, determine the maximum value of $a$.

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