
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 shopping mall installed an escalator between the first and second floors, moving upwards at a constant speed. A boy and a girl simultaneously walked up to the second floor on the escalator (the escalator was moving, and both of them were walking). If both of them walked at a constant speed, taking one step at a time, and the boy's step rate was twice that of the girl's. It is known that the boy took 27 steps to reach the top of the escalator, while the girl took 18 steps to reach the top.  
(1) How many steps of the escalator are exposed?  
(2) Now, next to the escalator, there is a staircase from the second floor down to the first floor, with the same number of steps as the escalator. After reaching the top of the escalator, both children walk down the staircase at their original speeds and then take the escalator up again (the distance between the escalator and the staircase is not considered). How many steps has the boy taken when he catches up with the girl for the first time?

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