
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 the function $f(x)=2\sin x\cos x+\cos 2x$ where $x\in \mathbb{R}$.

$(1)$ For what value of $x$ does the function $f(x)$ achieve its maximum value, and what is this maximum value?

$(2)$ If $\theta$ is an acute angle, and $f(\theta+ \frac {\pi}{8})= \frac { \sqrt {2}}{3}$, find the value of $\tan \theta$.

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