
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 function $f(x)$ is defined on the domain ${1, 2, 3, ..., 12}$ and satisfies the following conditions: ①$f(1)=1$; ②$|f(x+1)-f(x)|=1$ ($x=1,2,...,11$); ③$f(1)$, $f(6)$, $f(12)$ form a geometric sequence. The number of distinct functions $f(x)$ is $\_\_\_\_\_\_$.

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