
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 $\overrightarrow{a} = (\cos\theta, \sin\theta)$, $\overrightarrow{b} = (\cos\phi, \sin\phi)$  
(1) If $|\theta-\phi| = \frac{\pi}{3}$, find the value of $|\overrightarrow{a} - \overrightarrow{b}|$;  
(2) If $\theta+\phi = \frac{\pi}{3}$, let $f(\theta) = \overrightarrow{a} \cdot \overrightarrow{b} -\lambda|\overrightarrow{a} + \overrightarrow{b}|$, where $\theta \in [0, \frac{\pi}{2}]$. When $1 \leq \lambda \leq 2$, find the minimum value of $f(\theta)$.

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