
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 an arithmetic sequence ${a_n}$ where all terms are positive, the geometric mean of $a_2$ and $a_{12}$ is equal to $4$, determine the value of $a_3$ when $2a_5 + 8a_9$ is at its minimum.

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