
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 function $y=f(x)$ defined on the set $\{1,2,3,\ldots ,12\}$ satisfies the following conditions: (1) $f(1)=1$, (2) $|f(x+1)-f(x)|=1$ for $x=1,2,3,\ldots ,11$, and (3) $f(1)$, $f(6)$, $f(12)$ form a geometric sequence. The number of different functions $y=f(x)$ that satisfy these conditions is ______.

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