
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
For triangle $ABC$, sides are denoted as $a$, $b$, and $c$, with corresponding angles $\alpha$, $\beta$, and $\gamma$ such that $a < b < c$ and $\alpha < \beta < \gamma$. From an interior point $P$, perpendicularly drop lines to $a$, $b$, and $c$, meeting at points $D$, $E$, and $F$, respectively. Let $t = PD + PE + PF$. Express the maximum value of $t$ in terms of the altitudes $h_a$, $h_b$, and $h_c$.

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