
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 that $x > \frac{1}{3}$ and $y > 1$, if the inequality $\frac{9x^{2}}{a^{2}(y-1)} + \frac{y^{2}}{a^{2}(3x-1)} \geqslant 1$ holds true for any real numbers $x$ and $y$ that satisfy the conditions, find the maximum value of the real number $a$.

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