
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 $f(x)=2\sin x\cos x+2\cos ^{2}x-1$.

1. Find the value of $f(\frac{\pi}{8})$;
2. Find the monotonic intervals of the function $f(x)$;
3. If the equation $f(x)-a=0 (a\in R)$ has two distinct real roots $x_{1}$, $x_{2}$ in the interval $(0,\frac{\pi}{2})$, let $t=a\cos (x_{1}+x_{2})$, find the range of the real number $t$.

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