
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
Let \( P\left(x+a, y_{1}\right) \), \( Q\left(x, y_{2}\right) \), and \( R\left(2+a, y_{3}\right) \) be three distinct points on the graph of the inverse function of \( f(x)=2^x+a \). If \( y_{1} \), \( y_{2} \), and \( y_{3} \) form an arithmetic sequence with exactly one real value of \( x \), determine the range of values for the real number \( a \) and find the area of \( \triangle PQR \) when the distance from the origin to point \( R \) is minimized.

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