
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 vectors \(\boldsymbol{a}\), \(\boldsymbol{b}\), and \(\boldsymbol{c}\) such that
\[
|a|=|b|=3, |c|=4, \boldsymbol{a} \cdot \boldsymbol{b}=-\frac{7}{2}, 
\boldsymbol{a} \perp \boldsymbol{c}, \boldsymbol{b} \perp \boldsymbol{c}
\]

Find the minimum value of the expression
\[
|x \boldsymbol{a} + y \boldsymbol{b} + (1-x-y) \boldsymbol{c}|
\]
for real numbers \(x\) and \(y\).

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