
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
Let the three interior angles of $\triangle ABC$ be $A$, $B$, and $C$, respectively. Suppose the angle between $\overrightarrow {AB}$ and $\overrightarrow {BC}$ is $\theta$, given that $\overrightarrow {AB} \cdot \overrightarrow {BC} = 6$, and $6(2- \sqrt {3}) \leq |\overrightarrow {AB}||\overrightarrow {BC}|\sin(\pi-\theta) \leq 6 \sqrt {3}$.  
(Ⅰ) Find the value of $\tan 15^\circ$ and the range of values for $\theta$;  
(Ⅱ) Find the maximum value of the function $f(\theta) = \frac {1- \sqrt {2}\cos(2\theta- \frac {\pi}{4})}{\sin\theta}$.

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