
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 the diagram below, points $A$, $B$, $C$, and $Q$ are positioned such that $QA=3$, $QB=4$, $QC=5$, and $BC=6$. What is the maximum possible area of $\triangle ABC$?
[asy]
defaultpen(linewidth(0.8)); size(150);
pair B = (0,0), C = (6,0), A = (3,4), Q = (3.3,3);
draw(A--B--C--cycle^^B--Q^^C--Q^^A--Q);
label("$A$",A,N); label("$B$",B,S); label("$C$",C,S); label("$Q$",Q,S);
[/asy]

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