
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 $f(x)=A\sin (\omega x+\varphi)$ $(A > 0,\omega > 0,|\varphi| < \frac {\pi}{2})$ has the smallest positive period of $2$, and when $x= \frac {1}{3}$, $f(x)$ reaches its maximum value of $2$.  
$(1)$ Find the analytical expression of the function $f(x)$.  
$(2)$ Determine whether there exists an axis of symmetry for the graph of $f(x)$ in the closed interval $\left[ \frac {21}{4}, \frac {23}{4}\right]$. If it exists, find the equation of the axis of symmetry; if it does not exist, explain why.

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