
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 are 25 distinct prime numbers $p_{1}, p_{2}, \cdots, p_{25}$, each not exceeding 2004. Find the largest integer $T$ such that any integer whose digits sum to $T$ can be represented as the sum of distinct positive divisors of $\left(p_{1} p_{2} \cdots p_{25}\right)^{2}$. 

(Note: Examples such as $1 \cdot p_{1} + p_{1}^{2} + p_{1} p_{2} + p_{3}$ are sums of distinct positive divisors of $\left(p_{1} p_{2} \cdots p_{25}\right)^{2}$.)

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