
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

Find the maximum constant $\mathrm{c}$ such that for any positive integer $\mathrm{n}$ and any $\mathrm{n}$ positive real numbers $\mathrm{a}_{\mathrm{i}}$ $(1 \leq \mathrm{i} \leq \mathrm{n})$, the following inequality holds:

\[
\left(\sum_{i=1}^{n} a_{i}^{2}\right)\left(\sum_{1 \leq i, j \leq n} \max \{\mathrm{i}, \mathrm{j}\} \mathrm{a}_{\mathrm{i}} \mathrm{a}_{\mathrm{j}}\right) \geq c \cdot \left(\sum_{\mathrm{i}=1}^{n} \mathrm{a}_{\mathrm{i}}\right)^{4}
\]

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