
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 a rectangular box with dimensions $a \times b \times c$ where $a$, $b$, and $c$ are integers satisfying $1 \leq a \leq b \leq c$, the ratio of $a$ to $c$ is $1:4$, and the volume of the box is twice its surface area. Find the number of ordered triples $(a,b,c)$ that satisfy these conditions.

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