
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 that $\mathbf{Z}$ is the set of integers, define the sets $A=\{x \mid |x-3|<\pi, x \in \mathbf{Z}\}$, $B=\left\{x \mid x^{2}-11x+5<0, x \in \mathbf{Z}\right\}$, $C=\left\{x \mid 2x^{2}-11x+10\geqslant|3x-2|, x \in \mathbf{Z}\right\}$, and let $\bar{C}$ be the complement of $C$ in $\mathbf{Z}$. Find 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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