
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 proposition \( p \): there exists \( x_0 \in [0, 2] \), such that \( \log_2(x_0 + 2) < 2m \); proposition \( q \): the angle between the vectors \( \vec{a}=(1, m) \) and \( \vec{b}=(1, -3m) \) is acute.  
(I) If proposition \( q \) is true, find the range of the real number \( m \).  
(II) If the proposition \( (\neg p) \land q \) is true, find the range of the real number \( m \).

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