
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
$12.$ Given the function $f(x) = A\sin^{2}(\omega x+ \varphi )$ $(A > 0$, $\omega > 0$, $0 < \phi < \dfrac{\pi }{2} )$, where the maximum value of $y=f(x)$ is $2$, the distance between adjacent symmetry axes of its graph is $2$, and it passes through the point $(1,2)$.

$(1)$ Find $\phi$;

$(2)$ Calculate $f(1)+f(2)+⋯ +f(2017)$

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