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

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