
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 the three internal angles $A$, $B$, and $C$ of triangle $ABC$ are opposite sides $a$, $b$, and $c$ respectively. If $a\cos C+c\cos A=3$, and $a^{2}+c^{2}=9+ac$, then the maximum value of the area of triangle $\triangle ABC$ is ____.

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