
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 \( f(x) \) defined on the set \( \left\{ x \in \mathbf{Z}_{+} \mid 1 \leqslant x \leqslant 12 \right\} \) satisfies
\[ |f(x+1) - f(x)| = 1 \quad \text{for} \; x = 1, 2, \ldots, 11, \]
and \( f(1), f(6), f(12) \) form a geometric sequence. Given that \( f(1) = 1 \), find the number of different functions \( f(x) \) that satisfy these conditions.

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