
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 a function $f(x)=\sin (\omega x- \frac{\pi}{5})$ ($\omega > 0$) with the smallest positive period of $\pi$, after shifting the graph of $y=f(x)$ to the left by $\frac{\pi}{5}$ units, determine the symmetry properties of the graph of the resulting function $g(x)$.

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