
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
The  $\emph{Stooge sort}$  is a particularly inefficient recursive sorting algorithm defined as follows: given an array  $A$  of size  $n$ , we swap the first and last elements if they are out of order; we then (if  $n\ge3$ ) Stooge sort the first  $\lceil\tfrac{2n}3\rceil$  elements, then the last  $\lceil\tfrac{2n}3\rceil$ , then the first  $\lceil\tfrac{2n}3\rceil$  elements again. Given that this runs in  $O(n^\alpha)$ , where  $\alpha$  is minimal, find the value of  $(243/32)^\alpha$ .

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