
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 $\overrightarrow{a}=(\sin x, \cos x)$, $\overrightarrow{b}=(\sin x, \sin x)$, $\overrightarrow{c}=(-1, 0)$.  
(1) If $x= \frac{\pi}{3}$, determine the angle $\theta$ between vectors $\overrightarrow{a}$ and $\overrightarrow{c}$.  
(2) If $x \in \left[-\frac{3\pi}{8}, \frac{\pi}{4}\right]$, and the function $f(x)=\lambda \overrightarrow{a} \cdot \overrightarrow{b}$ has a maximum value of $\frac{1}{2}$, find the real number $\lambda$.

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