
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
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given that $$a-b= \frac {1}{3}c$$, $3\sin B=2\sin A$, $2\leq ac+c^2\leq32$, let the area of $\triangle ABC$ be $S$, and $$t= \frac {S+2 \sqrt {2}}{a}$$. Find the minimum value of $t$.

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