
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
Starting with ten billion, Barbara creates a sequence where she alternately divides by 5 and multiplies by 2. The sequence begins with 10,000,000,000. What is the last integer in her sequence after 12 steps, and how can this number be expressed in the form $a^b$, where $a$ and $b$ are positive integers, and $a$ is as small as possible?

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