
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
A sequence \(\left\{a_{n}\right\}_{n \geq 1}\) of positive reals is defined by the rule \(a_{n+1} a_{n-1}^{5} = a_{n}^{4} a_{n-2}^{2}\) for integers \(n > 2\) together with the initial values \(a_{1} = 8\), \(a_{2} = 64\), and \(a_{3} = 1024\). Compute
\[
\sqrt{a_{1}+\sqrt{a_{2}+\sqrt{a_{3}+\cdots}}}
\]

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