
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 the quadratic function \( y = x^2 + 2mx - 3m + 1 \), where the variable \( x \) and real numbers \( p \) and \( q \) satisfy:
\[ 4p^2 + 9q^2 = 2, \]
\[ \frac{1}{2}x + 3pq = 1, \]
and the minimum value of \( y \) is 1. Find the value of \( m \).

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