
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 $\vec{a}=(\cos x, \sin x)$, $\vec{b}=(\cos y, -\sin y)$, where $x \in [0, \pi]$

(1) Find $\vec{a} \cdot \vec{b}$ and $|\vec{a} + \vec{b}|$;  
(2) If $f(x) = \vec{a} \cdot \vec{b} - 2\lambda|\vec{a} + \vec{b}|$ has a minimum value of $-3$, find the value of $\lambda$.

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