
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 that the three vertices of triangle $\triangle ABC$ are $A(a, a+1)$, $B(a-1, 2a)$, and $C (1, 3)$, and all points inside and on the boundary of $\triangle ABC$ are within the region defined by $3x + y \geq 2$, determine the range of values for $a$.

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