
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 smallest positive constant  $c$  satisfying: For any simple graph  $G=G(V,E)$ , if  $|E|\geq c|V|$ , then  $G$  contains  $2$  cycles with no common vertex, and one of them contains a chord.

Note: The cycle of graph  $G(V,E)$  is a set of distinct vertices  ${v_1,v_2...,v_n}\subseteq V$ ,  $v_iv_{i+1}\in E$  for all  $1\leq i\leq n$   $(n\geq 3, v_{n+1}=v_1)$ ; a cycle containing a chord is the cycle  ${v_1,v_2...,v_n}$ , such that there exist  $i,j, 1< i-j< n-1$ , satisfying  $v_iv_j\in E$ .

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