
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
Determine an expression for the area of $\triangle PDB$ in terms of $h$ and $b$. Assume that $D(h, h)$ lies on the line $y = x$ and $h<b$. Coordinates of points are as follows: $P(0, b)$, $B(b, 0)$, and $D$. The coordinates for the relevant points are $P(0,b)$, $B(b,0)$, and $D$.
[asy]
size(5cm);defaultpen(fontsize(9));
pair p = (0, b); pair b = (b, 0); pair d = (h, h);
draw((-2, 0)--(15, 0), Arrow);
draw((0, -2)--(0, 15), Arrow);
draw(p--b--d--cycle);
label("$P(0, b)$", p, NW);
label("$B(b, 0)$", b, E);
label("$D(h, h)$", d, NE);
label("$x$", (15, 0), E);
label("$y$", (0, 15), N);
[/asy]

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