
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 PROBLEM OF PAINTING THE THÁP RÙA (THE CENTRAL TOWER) MODEL]
The following picture illustrates the model of the Tháp Rùa (the Central Tower) in Hanoi, which consists of  $3$  levels. For the first and second levels, each has  $10$  doorways among which  $3$  doorways are located at the front,  $3$  at the back,  $2$  on the right side and  $2$  on the left side. The top level of the tower model has no doorways. The front of the tower model is signified by a disk symbol on the top level.
We paint the tower model with three colors: Blue, Yellow and Brown by fulfilling the following requirements:
1. The top level is painted with only one color.
2. In the second level, the  $3$  doorways at the front are painted with the same color which is different from the one used for the center doorway at the back. Besides, any two adjacent doorways, including the pairs at the same corners, are painted with different colors.
3. For the first level, we apply the same rules as for the second level.
![Image](https://cdn.artofproblemsolving.com/attachments/2/3/18ee062b79693c4ccc26bf922a7f54e9f352ee.png)
(a) In how many ways the first level can be painted?
(b) In how many ways the whole tower model can be painted?

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