
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
Given $\mathbf{Z}$ as the set of integers, define the sets $A=\{x \mid | x-3 |<\pi, x \in \mathbf{Z}\}$, $B=\{x \mid x^{2}-11 x+5<0, x \in \mathbf{Z}\}$, $C=\{x\mid 2 x^{2}-11 x+10 \geq 3 x-2, x \in \mathbf{Z}\}$, and $\bar{C}$ as the complement of $C$ in $\mathbf{Z}$. Determine the number of proper subsets of $A \cap B \cap \bar{C}$.

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