
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 that point $O$ is the coordinate origin, and points $A(1,0)$, $B(0,1)$, $C(2\sin \theta,\cos \theta)$,
(1) If $| \overrightarrow{AC}|=| \overrightarrow{BC}|$, find the value of $\tan \theta$;
(2) If $( \overrightarrow{OA}+2 \overrightarrow{OB})\cdot \overrightarrow{OC}=1$, find the value of $\sin \theta\cos \theta$.

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