
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 $f(x) = \sin(\omega x + \varphi)$, where $\omega > 0$ and $|\varphi| < \frac{\pi}{2}$,
(1) If $\sin \frac{3\pi}{4}\sin\varphi - \cos \frac{\pi}{4}\cos\varphi = 0$, find the value of $\varphi$;
(2) Under the condition in (1), the distance between two adjacent axes of symmetry of the graph of $f(x)$ is $\frac{\pi}{3}$, determine the analytical expression of $f(x)$;
(3) Under the condition in (2), when the function $f(x)$ is translated $m$ units to the left and the resulting function is an even function, find the value of the smallest positive real number $m$.

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