
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
Suppose $\boldsymbol{a}_0$ is a unit vector. (1) If $\boldsymbol{a}$ is some vector in the plane, then $\boldsymbol{a} = |\boldsymbol{a}| \cdot \boldsymbol{a}_0$; (2) If $\boldsymbol{a}$ is parallel to $\boldsymbol{a}_0$, then $\boldsymbol{a} = |\boldsymbol{a}| \cdot \boldsymbol{a}_0$; (3) If $\boldsymbol{a}$ is parallel to $\boldsymbol{a}_0$ and $|\boldsymbol{a}| = 1$, then $\boldsymbol{a} = \boldsymbol{a}_0$. Among the above statements, the number of false propositions is ______.

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