
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 $\vec{a}=\{\sin 2 \varphi, 1, \tan \varphi\}$, $\vec{b}=\{1, \cos \varphi, \cot \varphi\}$ with $\varphi \in[0,2 \pi)$, and $\vec{a} \cdot \vec{b}=1$, find the values of $\varphi$.

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