
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 
$$
\begin{cases} 
x > \frac {1}{3} \\ 
y > 1
\end{cases},
$$
for any real numbers $x$ and $y$ that satisfy the conditions, the inequality 
$$
\frac {9x^{2}}{a^{2}(y-1)} + \frac {y^{2}}{a^{2}(3x-1)} \geq 1
$$ 
always holds. Determine the maximum value of the real number $a$.

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