
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, points $A$, $B$, $C$, and $P$ are positioned such that $PA=4$, $PB=5$, $PC=3$, and $BC=4$. What is the maximum possible area of $\triangle ABC$?
```asy
defaultpen(linewidth(0.8));
size(150);
pair B = (0,0), C = (4,0), A = (2,5), P = (2.2,2);
draw(A--B--C--cycle^^B--P^^C--P^^A--P);
label("$A$", A, N); 
label("$B$", B, S); 
label("$C$", C, S); 
label("$P$", P, S);
```

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