
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
To promote the construction of public fitness facilities, a sports center is planning to renovate and expand a sports field. Currently, two construction teams, Team A and Team B, are involved in the construction with the following information:<br/>Information 1<br/><table><tbody><tr><td width="66">Team</td><td width="236">Daily Construction Area (in $m^{2}$)</td><td width="190">Daily Construction Cost (in yuan)</td></tr><tr><td>A</td><td>$x+300$</td><td>$3600$</td></tr><tr><td>B</td><td>$x$</td><td>$2200$</td></tr></tbody></table>Information 2<br/><table><tbody><tr><td width="568">The number of days Team A needs to construct $1800m^{2}$ is equal to the number of days Team B needs to construct $1200m^{2}$.</td></tr></tbody></table>$(1)$ Find the value of $x$;<br/>$(2)$ The project is planned to be initially constructed by Team A alone for several days, followed by Team B continuing the construction alone. Both teams work for a total of $22$ days, and the completed construction area is not less than $15000m^{2}$. How much is the minimum construction cost that the sports center needs to pay during this period?

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