
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 $\overset{→}{a} =(\sin x,1)$ and $\overset{→}{b} =(1,k)$, and $f(x)=\overset{→}{a} \cdot \overset{→}{b}$.

(1) Find the range of the real number $k$ such that the equation $f(x)=1$ has a solution in terms of $x$.

(2) If $f(\alpha) = \frac{1}{3}+k$ and $\alpha \in (0,\pi)$, find $\tan \alpha$.

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