
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
There are two categorical variables $x$ and $y$, with one set of observations in a $2\times 2$ contingency table as follows:

|         | $y_{1}$ | $y_{2}$ | Total |
|---------|---------|---------|-------|
| $x_{1}$ | $a$     | $15-a$  | $15$  |
| $x_{2}$ | $20-a$  | $30+a$  | $50$  |
| Total   | $20$    | $45$    | $65$  |

where $a$ and $15-a$ are both integers greater than $5$. Find the value of $a$ such that, with a probability of error not exceeding $0.01$, it can be concluded that there is a relationship between $x$ and $y$.

Given: ${K^2}=\frac{{n{{({ad-bc})}^2}}}{{({a+b})({c+d})({a+c})({b+d})}}$

| $P(K^{2}\geq k)$ | $0.10$ | $0.05$ | $0.025$ | $0.010$ | $0.005$ |
|------------------|--------|--------|---------|---------|---------|
| $k$              | $2.706$| $3.841$| $5.024$ | $6.635$ | $7.879$|

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